HORIZONS OF DESCRIPTION:

BLACK HOLES AND COMPLEMENTARITY

A Dissertation

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Peter Joshua Martin Bokulich, B.A., M.A.

_____________________________Don A. Howard, Director

Graduate Program in Philosophy

Notre Dame, Indiana

April 2003

© Copyright by

PETER BOKULICH

2003

All rights reserved

HORIZONS OF DESCRIPTION:

BLACK HOLES AND COMPLEMENTARITY

Abstract

by

Peter Joshua Martin Bokulich

Niels Bohr famously argued that a consistent understanding of quantum

mechanics requires a new epistemic framework, which he named complementarity. This

position asserts that even in the context of quantum theory, classical concepts must be

used to understand and communicate measurement results. The apparent conflict

between certain classical descriptions is avoided by recognizing that their application now

crucially depends on the measurement context.

Recently it has been argued that a new form of complementarity can provide a

solution to the so-called information loss paradox. Stephen Hawking argues that the

evolution of black holes cannot be described by standard unitary quantum evolution,

because such evolution always preserves information, while the evaporation of a black

hole will imply that any information that fell into it is irrevocably lost – hence a

“paradox.” Some researchers in quantum gravity have argued that this paradox can be

resolved if one interprets certain seemingly incompatible descriptions of events around

black holes as instead being complementary.

Peter Joshua Martin Bokulich

In this dissertation I assess the extent to which this black hole complementarity

can be undergirded by Bohr's account of the limitations of classical concepts. I begin by

offering an interpretation of Bohr's complementarity and the role that it plays in his

philosophy of quantum theory. After clarifying the nature of classical concepts, I offer an

account of the limitations these concepts face, and argue that Bohr's appeal to disturbance

is best understood as referring to these conceptual limits.

Following preparatory chapters on issues in quantum field theory and black hole

mechanics, I offer an analysis of the information loss paradox and various responses to it.

I consider the three most prominent accounts of black hole complementarity and argue

that they fail to offer sufficient justification for the proposed incompatibility between

descriptions.

The lesson that emerges from this dissertation is that we have as much to learn

from the limitations facing our scientific descriptions as we do from the successes they

enjoy. Because all of our scientific theories offer at best limited, effective accounts of the

world, an important part of our interpretive efforts will be assessing the borders of these

domains of description.

ii

For my parents,

Paul and Pat Bokulich,

without whom this project could never have been started,

and for Alisa,

without whom it could never have been finished

iii

CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 1: BOHR’S COMPLEMENTARITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1 Complementary Quantum Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Measuring Quantum Electromagnetic Field Components . . . . . . . . . . . . . . . . . . . . 18

1.3 Classical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Measurement and Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.5 Uncontrollable Interactions and the Limits of Classical Concepts . . . . . . . . . . . . . . 46

CHAPTER 2: QUANTUM FIELD THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.1 Free QFT and Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.2 Interacting QFT and the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3 Unitary Equivalence and Algebraic QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.4 Microcausality and Complementarity in QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

iv

CHAPTER 3: BLACK HOLE THERMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.1 Holes and Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Black Hole Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3 Rindler Spacetime and Coordinate Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4 The Unruh Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6 The Generalized Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

CHAPTER 4: THE INFORMATION LOSS PARADOX . . . . . . . . . . . . . . . . . . . . . . . 109

4.1 Pure to Mixed State Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 Superscattering Matrices and Nonunitary Evolution . . . . . . . . . . . . . . . . . . . . . . . 115

4.3 Bleaching, Cloning, and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.4 Black Hole Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.5 The Limits of Local QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

CHAPTER 5: BLACK HOLE COMPLEMENTARITY . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1 ’t Hooft’s S-Matrix Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2 Incompatible Field Measurements in the S-Matrix Ansatz . . . . . . . . . . . . . . . . . . . 160

5.3 Nice Slices and Three Postulates of Black Hole Complementarity . . . . . . . . . . . . 173

5.4 Susskind, Thorlacius, and Uglum on Black Hole Complementarity . . . . . . . . . . . 183

5.5 Space-time Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

v

CHAPTER 6: EXPLANATORY POTENTIAL OF BLACK HOLECOMPLEMENTARITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.1 String Theory and the Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.2 Classical Concepts and Hidden Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.3 Complementarity and the Limits of Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 235

APPENDIX A: CONFORMAL STRUCTURE AND PENROSE DIAGRAMS . . . . . 244

APPENDIX B: MODELS OF BLACK HOLE COMPLEMENTARITY . . . . . . . . . . . 248

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

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FIGURES

Figure 1.1: The two-slit experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 1.2: A field measurement over regions I and II. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 3.1: Conformal spacetime diagram of black hole. . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 3.2: Rindler Spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 3.3: Outgoing mode in a black hole spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Figure 3.4: Laws of black hole mechanics and thermodynamics. . . . . . . . . . . . . . . . . . 102

Figure 4.1: Spacetime with evaporating black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 4.2: Banks’ residual black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 4.3: BER’s remnant scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 4.4: Giddings’ black hole remnant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 4.5: Adjusted diagram of remnant scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 4.6: Giddings’ Massive Remnant Proposal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 5.1: Shift of outgoing particle (’t Hooft). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 5.2: ’t Hooft’s spacetime for evaporating black hole. . . . . . . . . . . . . . . . . . . . . 158

Figure 5.3: Early- and late-time observers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure A.1: Penrose conformal spacetime diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Figure A.2: Einstein static spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

vii

ACKNOWLEDGEMENTS

I would like to begin by expressing my very deep gratitude to Jim Cushing who,

from the time I entered graduate school, tirelessly guided me through the mazes of

quantum mechanics and philosophy of physics. His lightning-quick wit and thunder-deep

insights had a profound influence on me; many aspects of this dissertation are a direct or

indirect outgrowth of my conversations with him. He was a brilliant scholar and a

wonderful man, and he is deeply missed.

My profound thanks also go out to Ernan McMullin who introduced me to the

subtleties of philosophy and history of science. I often learned more from one of his off-

the-cuff remarks during a reading group than I have from entire books. I would also like

to thank Jay Kennedy for his mentoring during my early years of graduate study and for

the hours he spent helping me work through various issues in the philosophy of

spacetime. His infectious enthusiasm for philosophy and physics at this early stage has

helped carry me through the remaining years.

I thank Samir Bose for his corrections and guidance on many of the physical

arguments discussed in this work and for many enlightening discussions on the topics

raised by them. I thank Ikaros Bigi not only for his inestimable help with this dissertation

but also for reaching out to offer classes to introduce philosophers like me to the exciting

viii

issues raised by the foundations of quantum field theory. It is efforts by physicists such

these that make an interdisciplinary project like this dissertation possible.

I owe a special thanks to Laura Ruetsche, who blazed a philosophic trail into

several of the areas of physics that this dissertation explores. Her work and her kind

guidance over the years have helped me to avoid several wrong turns and to escape many

pitfalls into which I had strayed.

My deepest intellectual debt, however, is owed to Don Howard. Without his

mastery of many diverse fields and his patient guidance, this project would never have

been conceived, let alone have been carried through to completion. It was he who

revealed to me the rich rewards of confronting philosophical and historical questions

together, as a means of better addressing both. I am extremely grateful for our many

discussions from which I learned more than I can say; I, along with this dissertation, have

benefitted enormously from his insightful lessons and critiques.

Finally, I would like to extend my thanks to my many family members and

friends, on whose love and support I have depended through the years. I would like

especially to acknowledge my parents, Pat and Paul Bokulich; they not only introduced

me to philosophy and nurtured me physically, spiritually, and intellectually, but they have

also been an unending source of encouragement and inspiration. And to Alisa, my wife,

my deepest love, and my most treasured intellectual companion: thank you.

1

INTRODUCTION

Looking back at the development of physics in the 20th century, one should be

impressed both by quantum theory’s stunning success in accounting for so many of the

processes we find in the world and the fact that it has been unable to offer an account of

one of nature’s most prominent forces: gravity. There are a number of reasons to believe

that the difficulties in developing a theory of quantum gravity are not of a purely technical

nature, but are an indication that a unification of general relativity and quantum theory

will require revisions in the conceptual foundations of one or both of these theories. By

studying attempts to formulate a theory of quantum gravity, one can therefore hope to

gain some insight into these conceptual foundations and also into the limitations that they

are likely to face.

A question that received a good deal of attention in the quantum gravity

community in the first half of the 1990s is whether we should expect the theory of

quantum gravity to be expressible in the standard quantum-mechanical framework of the

unitary evolution of states in a Hilbert space. In 1976 Hawking had proposed an

argument for the claim that the evolution and complete evaporation of a black hole could

not be described by such unitary evolution. While this argument was discussed

sporadically over the next couple of decades, it attracted considerably more attention

when certain two-dimensional models of black hole evolution fueled a hope for an

2

explicit theoretical model of such processes. A fairly sizable debate then developed over

the force of Hawking’s argument and its lessons for a theory of quantum gravity. Two

fundamental principles seemed to be in conflict: the so-called principle of causality,

which tells us that no signals can travel faster than light, and the principle of the unitary

evolution of states, which in quantum theory enforces a subtle balance between energy-

momentum conservation and locality.

It is common to claim that the unitary evolution of states represents the

conservation of information in all physical processes: if the evolution of a system is

unitary then, in principle, one could perform an appropriate complete set of

measurements on the system at some late time and infer the precise original state of the

system. The successes of quantum theory lead us to believe that in this sense no

information is lost, at a fundamental physical level, when one takes a book and burns it,

for example. In principle, as long as the entire system can be considered closed, it should

be possible to perform measurements on all the outgoing light and heat, on the ashes,

smoke, and surrounding air molecules, and reconstruct the precise initial state of the

system: The information is still there, it is merely in a highly scrambled form that is far

beyond our practical ability to reconstruct.

Suppose, however, that instead of burning the book, we throw it into a black hole.

Because nothing can escape from a black hole, once the book passes through the event

horizon, its information can never be returned to the external universe. This in itself

would not pose a problem for the conservation of information (i.e., for unitary evolution)

if the black hole were a permanent entity that could serve as a repository for the

3

information that fell into it. However, in 1974 Hawking established that black holes will

give off radiation and thus will lose mass-energy. This seems to imply that the black hole

will eventually disappear completely, and since the information from our book cannot

escape, when the black hole disappears the information will have to go with it. This

argument is referred to as Hawking’s information loss paradox.

While a number of responses have been offered to this argument, the one that has

gradually won the largest number of supporters goes by the name of black hole

complementarity. The advocates of this position point out that an external observer will

never see any objects, such as an infalling observer, cross the event horizon and enter the

interior of a black hole. Instead, the external observer will report a relativistic slowing of

all physical processes, such as the ticks of the infalling observer’s watch, near the event

horizon. The infalling objects will appear to freeze just above the event horizon for as

long as the black hole exists, and all the while the event horizon of the black hole should

be giving off heat that the external observer can measure. The suggestion of the

complementarians (i.e., the advocates of black hole complementarity) is that this outgoing

radiation carries off the information concerning any infalling objects in very much the

same way that the radiation resulting from burning the object would carry away the

information it originally contained. From the perspective of the external observer, no

information is lost in disappearing black holes because all infalling objects burn up at the

event horizon and re-emit their information in the outgoing Hawking radiation.

However, general relativity also tells us that the event horizon of a black hole is a

globally defined feature of a spacetime, a feature that should not generally be detectable

4

locally. This means that the infalling observer should not be able to tell when, or

whether, she is passing though the event horizon – which also implies that no quantum

gravitational effects should result in her destruction at the border of the black hole. The

complementarians accept this argument and claim that – from the infalling observer’s

perspective – neither she nor her books burn up at the event horizon of the black hole.

They then try to resolve the apparent contradiction between this claim and their previous

account of the external observer’s story by arguing that the descriptions offered by our

two observers are complementary, in the same sense that Bohr took descriptions of a

quantum object’s position and momentum to be complementary.

This novel appeal to complementarity comes when a number of philosophers are

taking a renewed interest in Bohr’s interpretation of the quantum formalism. There is

currently a small but growing movement in philosophy of physics that seeks to digest the

lessons presented by Bohr and apply them both to old problems, such as the EPR thought

experiment (Dickson 2003, Clifton and Halvorson 2002), and to new challenges, such as

those posed by algebraic quantum theory (Clifton 2000, Halvorson 2003). This

movement also questions the prevalent attitude that dismisses Bohr’s account as confused

or obscurantist (such claims are made, e.g., in Beller 1999) and tries to offer a more

nuanced account of Bohr’s position (see, for example, Howard 1994, 2003 and Tonona

2003).

This dissertation can be seen as a contribution to both aspects of this philosophical

movement. My primary aim is to assess the invocation of Bohr’s complementarity as a

potential resolution of the problem of applying quantum mechanics to black holes: Is the

5

scenario envisioned by the advocates of black hole complementarity actually analogous to

the cases considered by Bohr? Does these arguments offer a legitimate escape from the

paradox posed by Hawking? An answer to these questions will require an analysis of

Bohr’s complementarity, an account of the information loss paradox, and a study of

specific solutions to the paradox offered by the black hole complementarians. The

structure of the dissertation follows from considering each of these topics in turn.

Chapter 1 takes up the historical and philosophical challenge of explicating

Bohr’s account of complementarity. Bohr tells us that the quantum-mechanical

formalism is to be interpreted in terms of the complementary use of classical descriptions

to describe the outcomes of measurements and the functioning of measuring

arrangements, but there has been some controversy over what these classical descriptions

amount to. I argue that these descriptions are to be understood as the properties and laws

of our classical physical theories. My support for this reading comes from an

investigation of a widely cited but little understood paper by Bohr and Rosenfeld on the

measurability of quantum field values, and also from an analysis of the conceptual work

that Bohr believes classical concepts provide. After arguing that the concept of

disturbance plays an important and legitimate role in Bohr’s interpretation, I also consider

several misreadings of Bohr’s complementarity and argue that Bohr was never committed

to a form of operationalism or verificationism.

The next two chapters are largely expository. Chapter 2 offers an overview of

quantum field theory to lay the groundwork for later discussions of the information loss

paradox and black hole complementarity. Chapter 3 considers the descriptions of black

6

holes offered by general relativity and quantum field theory in curved spacetime. A key

topic here will be Hawking’s derivation predicting that black holes give off radiation like

a hot body, as well as further parallels between black holes and systems described by

thermodynamics.

Chapter 4 turns to the information loss paradox. While this discussion primarily

aims to set the stage for a discussion of black hole complementarity, it also addresses

several interpretive issues surrounding the debate over the paradox. I argue that an often

discussed, though never embraced, response to Hawking – namely the quantum cloning

response – is in fact a misguided suggestion that derives from an over-reification of the

concept of information. I also defend another response – that of black hole remnants –

from the charge that this proposal fails to meet adequately the challenge posed by

Hawking. These discussions serve to clarify the nature of the debate over information

loss as a debate over the appropriate limits of our effective theories.

Chapter 5 considers the three most prominent accounts of black hole

complementarity and evaluates the extent to which their claims can be seen as examples

of Bohr’s complementarity. I argue that none of these accounts offers a satisfactory

account of the incompatibility – required for a legitimate appeal to complementarity –

between the descriptions offered by the infalling and external observers. The account

offered by ’t Hooft seems to rest on an illegitimate appeal to measurement collapse to

secure this incompatibility; while that of Susskind and collaborators appeals to a form of

verificationism to rule out the joint descriptions of the two measurement outcomes.

However, I argue that neither of these moves is sanctioned by Bohr’s account of

7

complementarity. The final position considered is that of Kiem, Verlinde, and Verlinde,

whose account, I argue, postulates the correct structure to be considered a form of

complementarity, but offers little by way of justifying this structure. Thus, to the question

of whether black hole complementarity is actually complementarity, I offer the somewhat

cautious reply that while this possibility is left open by the considerations of Chapter 5,

none of the arguments offered by the complementarians succeeds in establishing that it

does in fact deserve to be considered an example of Bohr’s complementarity.

Chapter 6 concludes the dissertation by indicating some directions of current

research in quantum gravity that bear on the question of black hole complementarity, and

by offering an evaluation of the explanatory power of Bohr’s notion of complementarity.

Several results coming out of studies of superstring theory offer some speculative support

for the claim that the picture offered by the complementarians may accurately capture key

features of quantum gravitational processes. I argue that should these speculations turn

out to be accurate, complementarity may indeed play a key role in the escape from

Hawking’s paradox, despite the failure of the arguments considered in Chapter 5 to

establish the legitimacy of this move.

The final sections of the dissertation address Bohr’s arguments for the claim that

complementarity accurately captures the limits of our classical concepts. I argue that a

key aspect of Bohr’s account is the assumption that, when we are in a limit in which we

can apply the laws of our classical theory, the appropriate concepts to apply are precisely

those of that classical theory. This will imply, for example, that measurements that would

classically count as a position measurement will also count as position measurements in a

8

quantum world, so long as it is legitimate to describe the measurement process in

classical terms. Likewise, a momentum measurement will be a measurement of the actual

momentum of the quantum system. The fact that this assumption is denied by hidden

variable theories such as Bohm’s helps to illustrate the foundational interpretive role that

it plays in Bohr’s account. I conclude that Bohr’s is a consistent interpretation of the

quantum formalism, but that it is uncompelling in that it rests too heavily on interpreting

the properties of quantum objects in terms of effective classical theories.

1For ease of reference, citations of Bohr will be followed by the corresponding volumeand page number of Bohr’s Collected Works (BCW).

99

CHAPTER 1

BOHR’S COMPLEMENTARITY

Although complementarity was originally invoked in the context of interpreting

quantum theory, Bohr believed that this “viewpoint” had much broader epistemological

implications. In its most general form, complementarity is grounded in the fact that some

of our concepts or words require us to specify the context in which they can legitimately

be used: “The notion of complementarity . . . simply stresses the character of objective

description, independent of subjective judgment, in any field of experience where

unambiguous communication essentially involves regard to the circumstance in which

evidence is obtained” (Bohr 1961, p. 1105; BCW 10:405).1 Bohr suggests that while such

limitations are commonplace in fields such as psychology and sociology, it is a startling

lesson of the quantum revolution that these limitations also apply to our fundamental

physical theories.

The central focus of this chapter will not be this more general epistemological

formulation, but rather the more specific form of complementarity in which Bohr invokes

classical concepts to interpret the quantum formalism. Of particular interest will be the

“circumstances” that allow us to apply our concepts objectively, and the nature of the

1010

limitations – or “ambiguities” – facing our descriptions if we fail to specify these

contexts. It is this more specific quantum form of complementarity that black hole

complementarians appeal to in their attempts to avoid Hawking’s information loss

paradox; thus this chapter will establish the criteria that BHC (black hole

complementarity) will have to meet to be considered an example of Bohr’s

complementarity. We shall see in Chapter 5 that current arguments for BHC fail to meet

these criteria. However, if the position is viewed as a speculative proposal for the

features we might expect a full quantum theory to have, rather than as a compelling

argument for a new form of complementarity that invalidates the argument for the

nonunitary evolution of black holes, then BHC does offer a potential escape from

Hawking’s paradox.

Our aim in this chapter is limited to the explication of Bohr’s position, both

because this will lay the foundation for our later discussion and because offering an

accurate reading of Bohr’s position is itself a important conceptual challenge. We shall

return in Chapter 6 to the task of evaluating the position. At that point we shall briefly

consider an alternative interpretation of quantum theory, Bohm’s hidden-variable model,

as a way of clarifying key aspects of Bohr’s argument and to draw into question Bohr’s

claims that nature itself forces us accept his position on pain of inconsistency.

The structure of this chapter is as follows: The first section offers an overview of

complementarity and presents Bohr’s favorite illustration of its lessons, the two-slit

experiment. Section 1.2 steps through the basics of a largely neglected paper by Bohr and

Rosenfeld that offers a very detailed account of the possibilities of measuring quantum

1111

field values. The explicit nature of this example will be helpful in offering a concrete

interpretation of the broad and often vague account of complementarity offered by Bohr.

Section 1.3 is devoted to explicating the role of classical concepts in Bohr’s interpretation

and to defending my interpretation of these concepts in terms of classical physical

theories. The final two sections correct some common misreadings of Bohr and extract

key features of his account that will be important for evaluating BHC later in the

dissertation. Section 1.4 argues that a form of disturbance plays an essential role in

Bohr’s account of complementarity, but that he is not guilty of the problematic

conception of measurements disturbing otherwise well-defined properties of quantum

objects. Explicating the role of disturbance is important because, as we shall see in

Chapter 5, BHC fails to provide an adequate mechanism for this disturbance and thus

offers no justification for believing that it is an example of Bohr’s complementarity.

Section 1.5 differentiates Bohr’s view from operationalism; this too lays the groundwork

for our later evaluation of BHC.

1.1: Complementary Quantum Phenomena

In 1925 and 1926 the canonical formulation of quantum mechanics was put into

place. Bohr, however, considered the job unfinished, insofar as the physical content of

the theory still required clarification; an interpretation of the abstract theoretical

formalism still needed to be established. Whereas the formalism of classical mechanics

could be straightforwardly interpreted in terms of visualizable pictures, quantum

mechanics, according to Bohr, demanded a deep revision of our understanding of the

1212

relationship between our concepts and the world. At the 1927 Como conference Bohr put

forward his viewpoint of complementarity as the new conceptual framework for

interpreting the quantum formalism.

Most broadly, to say that two descriptions are complementary is to say that they

are jointly necessary but mutually exclusive. They are jointly necessary in that only

together do they exhaust the possible information that can be gathered about a system.

They are incompatible in that both descriptions cannot simultaneously be true of the

system.

In the context of quantum theory, one of the challenges in offering an account of

the relationship of complementarity is that of specifying the relata. Although Bohr often

speaks of complementary “concepts,” “words,” and “pictures,” his most considered view

is that the relationship should be applied to phenomena, by which he means “the

observations obtained under specified circumstances, including an account of the whole

experimental arrangement” (Bohr 1949, p. 238; BCW 7:378). Bohr goes on to tell us that

these conditions (or “specified circumstances”), which include an account of the

experimental arrangement, must be “defined by classical physical concepts” (ibid.). As I

shall argue below, by “classical concepts” Bohr means the properties and laws of classical

physical theories. However, the quantum nature of the world places limitations on the

applicability of these classical theories, and, therefore, on our ability to perform certain

kinds of measurements. The incompatibility of various experimental arrangements

implies that some phenomena will be mutually incompatible.

1313

Figure 1.1: The two-slit experiment.

The experiment to which Bohr most often appeals to illustrate complementarity is

the two-slit experiment. Figure 1.1 is based on Bohr’s (1949) illustrations in his

discussion of this experiment. The two-slit experiment consists of a particle incident

upon a diaphragm containing a slit of width a. The diaphragm can either be rigidly fixed

to a support that serves as a reference for our position measurements, or it can be free to

move in response to its interaction with the particle, thereby permitting momentum

measurements. If it is fixed, then we can infer the position of the particle in the vertical

direction at the time of the interaction with “uncertainty” )q = a. The key feature of this

arrangement is that we have correlated the position of the particle with the position of the

slit in the diaphragm at the time that the two interact. Because the diaphragm is rigidly

bolted to a lab bench, we can easily discover the position of the slit: We simply take a

ruler and measure the distance from the bench to the top and bottom of the slit. This is a

good example of what Bohr means when he tells us that the property of the measuring

1414

apparatus that is to be correlated with the property we are interested in measuring must be

“directly determinable according to its definition in everyday language or in the

terminology of classical physics” (Bohr 1939, p. 19; BCW 7: 311). We succeed in

measuring the particle’s position once we correlate it with the position of the slit, which

can be straightforwardly (and unambiguously) specified by a phrase such as “ten

centimeters above the lab bench.”

Now let us consider the case in which we wish to establish the momentum of the

particle after it passes through the slit. This requires us to allow the diaphragm to move

freely in response to its interaction with the particle. We can then measure the

momentum of the diaphragm before and after the particle passes through the slit, and

thereby assign a momentum to the particle and determine through which slit the particle

will subsequently pass. Measuring the momentum of the diaphragm is straightforward.

In Bohr’s words,

such measurements of momentum require only an unambiguous application of theclassical law of conservation of momentum, applied for instance to a collisionprocess between the diaphragm and some test body, the momentum of which issuitably controlled before and after the collision. It is true that such control willessentially depend on an examination of the space-time course of some process towhich the ideas of classical mechanics can be applied; if, however, all spatialdimensions and time intervals are taken sufficiently large, this involves clearly nolimitation as regards the accurate control of the momentum of the test bodies, butonly a renunciation as regards the accuracy of the control of their space-timecoordination. (Bohr 1935, 698; BCW 7:294)

This arrangement requires us to correlate the momentum of the particle with the

momentum of the diaphragm, which then in turn is correlated with the momentum of

some further test body. This final momentum is then measured (or “controlled”) once we

1515

can apply the “ideas of classical mechanics” to the “space-time course of some process.”

Bohr does not specify the sort of processes that are involved here, but it is clear that he

has in mind standard momentum measurements such as establishing the deflection of a

charged particle in a (classically described) magnetic field or observing how far the

particle can displace a (classically described) spring. The demand that the relevant

spacetime intervals be large, amounts to demanding that we be in the so-called classical

limit of quantum theory. We shall return to this point below in Section 1.3.

Bohr emphasizes that securing the applicability of the classical laws that allow us

to infer the momentum of the test bodies implies a “renunciation” of a similar classical

account of their positions, and therefore of the particle’s position. This is precisely

analogous to the fact that the use of the diaphragm as a position-measuring instrument

requires us to fix it rigidly to our lab bench, thus foregoing any chance of measuring its

momentum. In each case, our ability to describe one feature of the test bodies classically

demands that we allow a “latitude corresponding to the quantum-mechanical uncertainty

relations in our description” of the complementary feature (Bohr 1935, 698; BCW 7:294).

We are therefore faced with a choice as to whether we wish to measure the

position of a particle – by correlating its position with the position of a slit in a rigidly

fixed diaphragm – or to measure its momentum by freeing the diaphragm, correlating its

momentum with the momentum of the particle, and then measuring the diaphragm’s

momentum. The particle’s interaction with the diaphragm cannot be used to establish

2We should not think, however, that it is essential that the diaphragm be fixed if we wishto use it to measure the position of the particle. Bohr points out that we could begin witha free diaphragm with an accurately known momentum, and therefore an uncertainposition, and establish the position of the diaphragm only after the particle has passedthrough the slit. Thus even after the particle has passed through the free diaphragm, “weare after this passage left with a free choice whether we wish to know the momentum ofthe particle or its initial position” (Bohr 1935, p. 698; BCW 7:294). Such a delayedposition measurement is most easily accomplished by ensuring that the diaphragmoriginally has a negligible momentum relative to the rest of the apparatus. By increasingthe mass of the diaphragm we can, to any degree of accuracy desired, guarantee that thediaphragm will remain at rest even after it interacts with the particle and (potentially)gains some measurable momentum. Thus after the particle passes through we can, if wewish, simply bolt the diaphragm to a rigid support and measure the position of the slit,rather than measuring its momentum. While Bohr does not explicitly draw thisconclusion, it is helpful to see that this procedure would, for example, allow us toreconstruct the two-slit interference pattern even though the first slit was not rigidly fixedat the time the particle passed through it. We would simply have to adjust our recordingof each spot on the screen to account for the measured displacement of the slit in the firstdiaphragm.

1616

both the position and the momentum of particle – except within the limits set by the

uncertainty relations.2

As we shall discuss in more detail in Sections 1.4 and 1.5, the road from the

incompatibility of measurements to the incompatibility of concepts is a perilous one, on

which many interpreters of Bohr have gone astray. The story is complicated by the fact

that Bohr modified his presentation of this connection in response to the EPR (Einstein,

Podolsky, and Rosen) paper of 1935. Before this time he would speak of observations

requiring “an interference with the course of the phenomena” (Bohr 1929 p. 115;

BCW 6:249), and this interference was seen to play an important role in accounting for

the limitations of our classical concepts. After 1935, however, Bohr draws the

connection between the possibilities of measurement and the possibilities of applying

concepts through the adoption of a more technical and holistic account of phenomena. A

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phenomenon should not merely be considered a spot on a screen, a flash on a

spinthariscope, or a click of a counter. Instead, every well-defined quantum phenomenon

must make essential reference to the entire measurement context in which the observation

occurs:

[T]he word phenomenon [should be used] exclusively to refer to the observationsobtained under specified circumstances, including an account of the wholeexperimental arrangement. In such terminology, the observational problem is freeof any special intricacy since, in actual experiments, all observations areexpressed by unambiguous statements referring, for instance, to the registration ofthe point at which an electron arrives at a photographic plate. (Bohr 1949, p. 238;BCW 7:378)

In classical physics it is possible to abstract away from the context in which a particular

property value is obtained. This is because we can, in principle, offer a precise account of

how our instrument influenced, or contributed to, the particular effect we observed; and

thus we can consider the properties belonging solely to the object under consideration. In

a world governed by quantum mechanics, however, this is not possible. Quantum

phenomena, according to Bohr, are holistic and cannot be further analyzed:

The very fact that quantum phenomena cannot be analysed on classical lines thusimplies the impossibility of separating a behaviour of atomic objects from theinteraction of these objects with the measuring instruments which serve to specifythe conditions under which the phenomena appear. (Bohr 1948, p. 313; BCW7:331)

The holistic nature of the phenomena under investigation undergirds the incompatibility

of our complementary classical descriptions. It generally takes a carefully arranged

instrument to measure a particular property – i.e., to infer the value of a property from

some observation –, and this measurement arrangement will be incompatible with an

arrangement that would be required to measure some property represented by a quantum-

1818

mechanical operator that fails to commute with the operator representing the first

property.

A measurement allows us to attribute properties according to their classical

definition, but quantum mechanics implies fundamental limitations on our abilities to

make such attributions. Instrument arrangements intended to measure properties

represented by noncommuting operators will be incompatible. However, a single

classical description will not exhaust the possibilities of describing an object, for in other

contexts other property attributions would be possible. Thus certain classically described

phenomena are both mutually incompatible and jointly necessary for a complete

description; i.e., they are complementary.

1.2: Measuring Quantum Electromagnetic Field Components

Bohr’s treatment of quantum electromagnetic fields is particularly significant

because it forces him to develop the details of classically describable measuring

arrangements in far more detail than the more straightforward example of the double-slit

experiment. It also offers us an account of complementary descriptions besides those of

position and momentum, which offers important insight on how classical concepts are to

be employed.

Bohr and Rosenfeld’s 1933 paper addresses whether it is possible to test the newly

developed quantum electromagnetic formalism, or whether its specifically quantum

3I refer to the theory as quantum electromagnetism because it does not include an accountof the dynamics of the sources; i.e., it is source-free quantum electrodynamics.

1919

mechanical effects will be hidden by uncertainties.3 Landau and Peierls (1931) argued

that the quantum nature of fields cannot be measured because the radiation due to the

acceleration of the test body will introduce uncertainties that will prevent sufficiently

accurate measurements of the field quantities. Bohr and Rosenfeld respond by

demonstrating that as long as one considers the average field values over finite

spatiotemporal regions, as opposed to considering the values at a point, then one can

measure any single field component to any accuracy desired. Further, they demonstrate

that any two average field components that are represented by commuting operators can

be comeasured with arbitrary accuracy, but when we consider components whose

operators do not commute, the accurate measurement of one component will prevent the

precise determination of the other, as would be expected from the uncertainty relations

derivable from quantum electromagnetism.

It is helpful that Bohr and Rosenfeld begin their project by explaining what is

required to perform a measurement, both in the standard quantum-mechanical case and in

the case of quantum fields.

Characteristic of the [usual quantum-mechanical measurement] problem is thepossibility of attributing to each individual measurement result a well-definedmeaning in the sense of classical mechanics, while the quantum-imposedinteraction, uncontrollable in principle, between instrument and object is fullytaken into account through the influence of each measuring process on thestatistical expectations testable in succeeding measurements. (Bohr and Rosenfeld1933, p. 359; BCW 7:125)

4Note that this is not the standard modern account of the measurement problem, whichfocuses on the impossibility of unitarily evolving from a pure state to a mixture. We shallcontrast Bohr’s account with this standard version below in Section 1.3.

2020

Here we have one of the clearer statements of what Bohr believes is involved in the

“measurement problem.”4 The problem involves providing “meaning in the sense of

classical mechanics” (Deutung im Sinne der klassischen Mechanik) for each individual

measurement result, and the essentially quantum aspect of situation is accounted for by

recognizing that we can generally only make statistical statements about future

measurement results.

Bohr and Rosenfeld describe the further challenges that arise in the case of

quantum fields:

In contrast, in measurements of field quantities, indeed, every measuring result iswell defined on the basis of the classical field concept; but, the limitedapplicability of classical field theory to the description of the unavoidableelectromagnetic field effects of the test bodies during the measurement implies, aswe shall see, that these field effects to a certain degree influence the measurementresult itself in a way which cannot be compensated for. However, a closerinvestigation of the fundamentally statistical character of the consequences of thequantum-electromagnetic formalism shows that this influence on the object ofmeasurement by the measuring process in no way impairs the possibility of testingsuch consequences, but rather is to be regarded as an essential feature of theintimate adaptation of quantum field theory to the measurability problem. (ibid.)

Our interpretation of individual measurement results will again rely on classical concepts,

here classical electromagnetic theory, but now we shall find that the quantum-statistical

effects are not always limited to future measurements, but can also play a role in a single

measuring arrangement. The issue here is whether measurements of average field values

– possibly of different field components, and possibly over different spacetime regions –

2121

will interfere with one another, thus preventing us from attributing a value to the field on

the basis of our measurement outcome.

Quantum electromagnetism enables us to derive uncertainty relations for various

field components averaged over various regions of spacetime from the commutation

relations defined in the theory. Ideally we would like to show that we can measure two

average field values arbitrarily accurately whenever the commutator between these

averaged components vanishes. However, when the commutator is nonzero we should

find that the accuracy of our measurements is limited by the unavoidable interference of

one measurement process on the other – and this limitation on the possibility of such joint

measurements should agree in order of magnitude with the appropriate uncertainty

relation. The purpose of Bohr and Rosenfeld’s paper is to demonstrate that this is in fact

the case.

Let us look at the core of a relatively straightforward example to see how this

works. How do we go about measuring the components of a quantum electromagnetic

field? Precisely the same way we go about measuring the components of a classical EM

field, by using charged test bodies:

The measurement of electromagnetic field quantities rests by definition on thetransfer of momentum to suitable electric or magnetic test bodies. (Bohr andRosenfeld 1933, p. 368; BCW 7:134)

If we consider a test body that occupies some volume V, and that has a uniform charge

density D, then the momentum transferred to this body by the x-component of the electric

field in the period of time T is given by

pxO! pxN = D Ex V T. (1.1)

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As Bohr and Rosenfeld tell us in the above extract, this equation of classical

electrodynamics is our definition of the average electric field value Ex. Therefore, this

must serve as the foundation for an account of a quantum field component as well. If we

claim to have measured a value of the x-component of the quantum electric field, the very

meaning of this statement implies that we have measured the momentum of the test body

at two times, and have reliably been able to invoke Equation (1.1) to infer the value of Ex.

Now we need to consider the conditions under which (1.1) is valid, that is, the

conditions that will have to hold whenever we measure the value of Ex. First, we can only

invoke this equation if we are able to neglect the acceleration of the test body during the

time interval T. We can accomplish this, to any desired degree of accuracy, by using a

sufficiently massive test body. We also need to consider the limitations imposed by the

uncertainty relations on the state of the test body. Because we want to be able to perform

an arbitrarily accurate measurement of Ex, we require )px to be small, which implies an

uncertainty in x. However, this does not imply a fundamental limitation on the accuracy

with which we can measure Ex, for by increasing our charge density, D, we can still make

our uncertainty in the value of Ex as small as we please. We also need to be concerned

about the changes in the field values due to the acceleration of the test body during the

initial and final momentum measurements. These changes will be proportional to the

duration of the momentum measurements )t and therefore can be neglected if the ratio of

)t to T is sufficiently small.

The next effect that we will have to take account of is that of the change of the

field due to the charge of the test body. Classically we can neglect these effects because

2323

we can consider the limit of vanishingly charged test body of correspondingly small mass

and size. When measuring quantum fields, however, we cannot allow the charge density

to become arbitrarily small if we are to be able to measure the field values we are

interested in. We can compensate for the charge of the test body when it is at a fixed

position by arranging another body of equal and opposite charge density (!D) to cancel

the test body’s charge. However, the test body will have to be free if we are to measure

the momentum imparted to it by the field. Further, the initial momentum measurement

will imply that we lose our knowledge of the x-position of the test body. We can,

however, perform our momentum measurement in such a way that the test body will be

stationary after being displaced some unknown distance Dx. In Bohr and Rosenfeld’s

words,

outside the short time intervals occupied by the momentum measurements all thetest bodies employed in the field measurement can be considered as being at rest,which greatly simplifies the calculation. For immediately after each momentummeasurement, i.e. practically speaking still inside the interval )t, we can give thetest body system a second push in the opposite direction by means of a suitabledevice, such that the velocity change which every component body suffered in thefirst measurement is cancelled out; this can be done with arbitrary accuracy. . . .However, with this arrangement it is impossible to know the time intervalbetween the two collision processes with a latitude smaller than )t, and so, asrequired by the indeterminacy principle, the test body is not returned to its originalposition by the counter-collision, but rather is brought to rest to the requiredapproximation at an unknown position, displaced by a distance of the order )x.(Bohr and Rosenfeld 1933, p. 378; BCW 7:144)

Given this displacement of the body, we can calculate the classical changes in the field

due to the now non-zero charge distribution of the test body and its neutralizing

counterpart, which is still fixed to the common support. The value of this contribution to

the field component, averaged over some spatiotemporal region VII TII is given by

5 For those who feel more comfortable having the full expression in front of them, it isAxx

I,II = ! (1/VIVIITITII) ITI dt2 ITII dt2 IVI dv1 IVII dv2 [M:M< *(x2 ! x1)/r]

where *(x2 !x1) is the four-dimensional Dirac delta function, and r is the spatial distancebetween the two points. Regions I and II are lightlike related if and only if there exists apoint in I that is lightlike related to some point in II. If all points in I are spacelike(respectively timelike) related to all points in II, we say that I and II are spacelike(timelike) related.

2424

ExI, II = Dx D VI T Axx

I, II. (1.2)

Here VI is the volume occupied by our test body that is displaced by distance Dx during

the period T, and AxxI, II is a volume integral over a two-point function: The only feature of

AxxI,II that is significant for our discussion here is that it is nonzero only when regions I

and II are lightlike related; thus whenever two regions are spacelike or timelike related

AxxI,II equals zero.5

The key point here is that we cannot predict the precise value of this contribution

to the field because we do not know what the displacement Dx is – and it cannot be

measured without ruining our knowledge of the momentum of the test body, and hence

our measurement of the field value. However, we can see that the force that will be

exerted on a test body due to Ex will be proportional to this unknown displacement Dx.

Thus, if we are only interested in measuring a single average field value, we can

compensate for this effect by installing a spring that is appropriately calibrated to cancel

out the force exerted by ExI,I. This arrangement allows us to use Equation (1.1) to infer

the average field component Ex from our measurements of the test body’s momentum

The only remaining point of justification to address is the fact that we have merely

calculated the classical field effects of the displaced test body in (1.2) and have here

ignored the quantum nature of the field. However, the quantum nature of the field does

2525

not impose any additional limitations on the accuracy of our measurement. The value of

any field component will fluctuate around a classically described value unless we have

actually performed a measurement of that particular field value. That is, if we calculate

field values from either information about the quanta of the field, or from information

about the classical sources of the field, then there will be uncertainties in the field values,

or as Bohr and Rosenfeld put it, all predictions of average field values “must be of an

essentially statistical nature” (p. 387; BCW 7:153).

Further, the magnitude of the fluctuations is the same whenever the field is a

result of classically describable charges; it is precisely given by the equation for vacuum

field fluctuations. Thus our classically describable charge distribution will not introduce

any additional quantum effects to obscure our measurement: once we have compensated

for the classical value of the change in the field due to the movement of the test body,

there is no need to correct further the measurement results to account for the quantum

nature of the field. Bohr and Rosenfeld therefore conclude that we have accomplished

our task:

Without further correction, the measurement results obtained by means of theexperimental arrangement described thus appear as the desired field averages fortesting the theoretical statements. Such a view of measuring results . . . is alsosuggested by the fact that all measurements of physical quantities, by definition,must be a matter of the application of classical concepts; and that, therefore, infield measurements any consideration of limitations on the strict applicability ofclassical electrodynamics would be in contradiction with the measurementconcept itself. (Bohr and Rosenfeld 1933, p. 387; BCW 7:153)

Notice their reiteration of the fact that it is only insofar as we can apply the laws of

classical physics that we can claim to have performed a measurement.

2626

Figure 1.2: A field measurement over regions.I and II.

We have shown that it is possible to measure a single average field component,

but Bohr and Rosenfeld also want to demonstrate that field components of perhaps

distinct regions can be comeasured no more accurately than the limits set by the quantum

electromagnetic uncertainty relations. We saw above that the electric field due to a

displacement of a charged test body is given by (1.2). If we intend to measure the x-

component of the electric field over two spatiotemporal regions, I and II, then our

classical equations for the transfer of momentum to our first test body will be given by

pxI ! px

I = DI VI TI (Ex + ExI, I + Ex

II, I),

and a corresponding equation will hold for the second test body. We can once again

compensate for ExI, I with a properly calibrated spring, but now we must also somehow

account for the effect of the charge of the other test body, given by ExII, I.

To compensate for this field effect to the greatest degree possible, we take a third

body, this one uncharged, that will bridge the gap between our two test bodies. We attach

this uncharged rod by an appropriately calibrated spring to the first test body, as indicated

in Figure 1.2. We measure the momentum of the rod at the same time that we perform

2727

our initial and final momentum measurements on our first test body, that is, at the

beginning and end of TI. Immediately before performing these momentum

measurements, however, we send a light signal from our second test body to the rod, to

establish the relative positions of these two bodies.

Using our knowledge of the momenta of all three bodies and the relative position

of the rod and the second test body, we can account for all of the effects of the ExII, I term

(and the spring) on the first test body, except for a factor that is proportional to

DxII(Axx

I, II !AxxII, I). Likewise, the only uncertainty in how much momentum is transferred

to the second test body by the original field comes from a factor that is proportional to

DxI(Axx

I, II !AxxII, I). An explicit calculation of the magnitude of the resulting uncertainties

in the average field values yields

)ExI )Ex

II ~ S |AxxI, II !Axx

II, I|,

which is precisely the uncertainty relation derivable from the quantum electrodynamical

formalism.

Note that the limitations expressed by the uncertainty relations here come from

our inability to determine the extent to which the momentum transferred to the test bodies

comes from, e.g., ExI or Ex

II, I. If the contributions from ExII, I and Ex

I, II can be made to

vanish – for example, if region I and region II coincide – then we can measure the average

field components ExI and Ex

II as accurately as we please. But when we have nonzero

contributions to the field that are proportional to the unknowable displacements DxI and

DxII, we can only infer the average field values from our momentum measurements with

an accuracy given by the uncertainty relations.

2828

A measurement of an average value of a component of the quantum electro-

magnetic field requires the legitimate application of the laws of classical electro-

magnetism. A careful analysis of the possibility of applying these laws reveals that the

conditions required for precisely measuring one average field value are incompatible with

the conditions required for a precise measurement of other average values. The degree of

this incompatibility is given by the magnitude of the commutator of the operators

representing these average field values.

1.3 Classical Concepts

Now that we have in hand the concrete examples of the two-slit experiment and

measurements of average electromagnetic field components, we are ready to explicate the

meaning, function, and scope of classical concepts in Bohr’s theory. By classical

concepts Bohr means the properties and laws of classical physical theories such as

Newton’s mechanics and Maxwell’s electrodynamics. This interpretation is somewhat

controversial. For example, Don Howard (1994) has argued that Bohr’s notion of

classical concepts is best interpreted as referring to appropriate quantum mixed states.

While the true quantum description of the joint system composed of the object and the

apparatus is generally an inseparable entangled state, the mixed state is separable in that

it allows us to assign an independent state to each component. Such mixtures are

“appropriate” if they offer the same outcome statistics as the pure quantum state (which is

presumably the true state of the system). Although no single mixture can reproduce the

probabilities of a pure state for all possible measurements, for any given measurement

2929

there will be some mixture that will be appropriate in this sense. While this reading of

classical concepts in terms of appropriate mixtures does capture important aspects of

Bohr’s view – and while such mixtures can generally be reconstructed from the more

classical account that I am advocating – I argue that Howard’s view does not fully do

justice to the textual evidence or, perhaps more importantly, to Bohr’s actual use of

classical concepts when analyzing measurements.

Support for interpreting Bohr’s classical concepts as being descriptions that utilize

the properties and laws of classical physics comes from four corners: First and foremost,

it is supported by Bohr’s actual examples of measurement. Second, it makes sense of

Bohr’s insistence that not only our measurement outcomes, but also the functioning of our

measuring instruments must be described classically. Third, it explains the weight that

Bohr places on the correspondence principle and his comments regarding our freedom in

locating the divide between our quantum and classical descriptions. Fourth and finally, I

argue that only this reading can adequately account for the interpretive work that Bohr

demands of classical concepts. The remainder of this section will consider each of these

four points in turn, as this shall also serve to clarify the role that classical concepts play in

Bohr’s philosophy.

As we saw in the previous section, when Bohr sets out to measure average

quantum field values, he looks for the possibilities of applying the appropriate equation of

classical electromagnetism. The very definition of an average electromagnetic field

value, he claims, is given by the classical equation pxO! pxN = D Ex V T (Equation 1.1).

3030

Therefore, a measurement of such a value can only mean that we have legitimately

applied this equation to our measuring arrangement:

[A]ll measurements of physical quantities, by definition, must be a matter of theapplication of classical concepts; and . . . therefore, in field measurements anyconsideration of limitations on the strict applicability of classical electrodynamicswould be in contradiction with the measurement concept itself. (Bohr andRosenfeld 1933, p. 387; BCW 7:153)

The use of classical electromagnetism to infer a field value from our measurements of the

momentum transferred to our test bodies is a clear example of how Bohr intends for us to

analyze our measurement outcomes in terms of the properties and laws of classical

physics. Similarly, as we shall see more clearly below, Bohr’s prescriptions for applying

the concept of momentum also rely upon the applicability of classical laws such as the

momentum of a particle being equal to its mass multiplied by its velocity (where this last

might be inferred from the time it takes the particle to travel some known distance).

We turn to our second corner by asking what, according to Bohr, needs to be

described by classical concepts. Here he is clear that we must have a classical description

of both the outcomes of our experiments and also of the experimental arrangement: “in

every account of physical experience one must describe both experimental conditions and

observations by the same means of communication as one used in classical physics”

(Bohr 1958, p. 169; BCW 7:417). Thus we need a classical account not only of the

properties of interest (e.g., the momenta of our charged test bodies), but also of how these

bodies operate; that is, “the functioning of the measuring instruments must be described

within the framework of classical physical ideas” (Bohr 1958, p. 170; BCW 7:418,

emphasis added). The “classical physical ideas” that allowed us to describe the

3131

functioning of our charged test bodies were, of course, simply the laws expressed in

Equation (1.1).

Likewise, Bohr insists that to apply the concepts of momentum or energy, even in

the context of quantum mechanics, we must be able to use classical theory such as

Newtonian mechanics to analyze our measurements:

Strictly speaking, every reference to dynamical concepts implies a classicalmechanical analysis of physical evidence which ultimately rests on the recordingof space-time coincidences. Thus, also in the description of atomic phenomena,use of momentum and energy variables for the specification of initial conditionsand final observations refers implicitly to such analysis and therefore demandsthat the experimental arrangements used for the purpose have spatial dimensionsand operate with time intervals sufficiently large to permit the neglect of thereciprocal indeterminacy expressed by [)qA)p=h/4B]. (Bohr 1939, p. 315;BCW 7:333, emphasis added)

Bohr here is telling us that every time we refer to momentum or energy we must “strictly

speaking” be referring to a situation in which we can analyze some spatio-temporal

recordings using classical mechanics. Such an analysis is, of course, a classical account

of the functioning of the apparatus. Our ability to offer an account of the functioning of

the measuring instrument in terms of classical physics, when in fact both the instrument

and the object are ultimately quantum mechanical systems, rests on the existence of a

classical limit for quantum mechanics.

This brings us to the third category of support for reading Bohr’s classical

concepts as referring to the properties and laws of classical physical theories: only this

reading of classical concepts makes sense of Bohr’s repeated appeal to the classical limit

of quantum mechanics when explaining our ability to employ classical concepts to

describe measurement arrangements and outcomes. In the above extract, for example,

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Bohr tells us that the experimental arrangements that allow us to use the classical

concepts of momentum and energy must utilize spatiotemporal distances “sufficiently

large to permit the neglect” of the Heisenberg uncertainty relations. This is just to say

that the actions involved must be large enough to place us securely in the classical limit

of quantum theory; that is, the classical description of the functioning of the apparatus

will be effectively equivalent to the quantum description.

This same point can be seen in Bohr’s accounts of how classical concepts come

into play in momentum measurements. He tells us that “such measurements of

momentum require only an unambiguous application of the classical law of conservation

of momentum” (Bohr 1935, 698; BCW 7:294). However, merely transferring momentum

does not allow us to quantify it: asserting that a quantity of momentum has been

transferred from a particle to the diaphragm to some further test body does not yet tell us

how much momentum the particle possessed. This further analysis requires us establish

the momentum of the test body by subjecting it to a measurement procedure that can be

treated classically. As Bohr tells us, the final analysis of momentum

will essentially depend on an examination of the space-time course of someprocess to which the ideas of classical mechanics can be applied; if, however, allspatial dimensions and time intervals are taken sufficiently large, this involvesclearly no limitation as regards the accurate control of the momentum of the testbodies. (Bohr 1935, 698; BCW 7:294)

A description based on space-time pictures is precisely what classical mechanics offers

us. In his Como lecture Bohr tells us that such classical pictures can be employed only in

the classical limit of quantum mechanics, i.e., when the geometrical optics approximation

is valid:

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The limitation in the classical concepts expressed through [the uncertaintyrelations], is, besides, closely connected with the limited validity of classicalmechanics, which in the wave theory of matter corresponds to the geometricaloptics in which the propagation of waves is depicted through ‘rays’. Only in thislimit can energy and momentum be unambiguously defined on the basis of space-time pictures. (Bohr 1928 p. 582; BCW 6:150)

It is significant that the limits of our classical descriptions are tied to limits of the

applicability of geometrical optics, which is to say that the concepts can be safely used

only in the classical limit where the relevant actions are large compared to Planck’s

constant. It is also important that in each experimental test we are only concerned with

some finite degree of accuracy and to this degree of accuracy we can show that

some apparatus will admit of a sufficiently accurate classical description. Such a degree

of accuracy might be secured, for example, by considering a test body with a sufficiently

large mass and/or charge.

It is worth pausing to emphasize that Bohr is consistently interested only in some

finite degree of precision for our descriptions of measurement results. His argument

strategy is always that one is first to specify the finite degree of precision desired from a

measurement, and then one is to confront the problem of designing a measuring

arrangement that can reveal the property value to that degree of precision. This fact

undermines interpretations of Bohr’s complementarity that demand absolutely precise

(point-like) values of properties revealed in measurement. Halvorson (2003), for

example, argues that because Bohr’s “complementarity principle” requires there to be

absolutely sharp values of either position or momentum, we should view the

complementary descriptions in question to be the representations of the algebra of

6See Section 2.3 for a discussion of unitarily equivalent and inequivalent representationsof the algebra of quantum observables.

3434

observables that will make such precise value attributions possible. Because these

representations will be inequivalent,6 we can (according to Halvorson) understand Bohr’s

otherwise cryptic claims regarding the incompatibility of descriptions as referring to these

representations. While Halvorson’s insights may prove to be a helpful new extension of

complementarity, it clearly does not capture Bohr’s actual views – precisely because Bohr

was never interested in the possibility of attributing values with point-like precision.

Such values he would view as idealizations that need not be recovered with exactitude

from our theory.

Bohr places a great deal of interpretive weight on the fact that our measuring

instruments must be described classically although the quantum phenomena we are

interested in, such as particle interference patterns and the Compton effect, cannot be

explained by classical physics. We must therefore make a significant distinction between

the instrument, which must be described classically, and the object, which cannot be so

described. Indeed, he refers to this as the “principal distinction between classical and

quantum-mechanical phenomena” (Bohr 1935, p. 701; BCW 7:297, emphasis original),

and argues that the importance of this distinction lies in the fact that classical concepts

(which he seems to say are essentially tied to classical theories) – though clearly not

universally valid – must be used to interpret all measurements:

While . . . in classical physics the distinction between object and measuringagencies does not entail any difference in the character of the description of thephenomena concerned, its fundamental importance in quantum theory . . . has itsroot in the indispensable use of classical concepts in the interpretation of all

3535

proper measurements, even though the classical theories do not suffice inaccounting for the new types of regularities with which we are concerned inatomic physics. (Bohr 1935, p. 701; BCW 7:297)

Precisely where we place this divide between the classical and quantum descriptions in

this classically describable region has little significance, so long as it is placed somewhere

in the “region where the quantum-mechanical description of the process concerned is

effectively equivalent with the classical description” (Bohr 1935, p. 701; BCW 7:297).

This characterization of the distinction between classical and quantum

descriptions seems to imply that the former are based on dynamical laws that are

recovered in some “region” of quantum theory. This area of overlap is precisely the

classical limit, in which all relevant actions are large and one can interpret the behavior of

our systems in terms of the visualizable spacetime pictures of classical mechanics:

[I]t is, of course, to a certain degree a matter of convenience to what extent theclassical aspects of the phenomena are included in the proper quantum-mechanical treatment where a distinction in principle is made between measuringinstruments, the description of which must always be based on space-timepictures, and objects under investigation, about which observable predictions canin general only be derived by the non-visualizable formalism. (Bohr 1948 p. 315;BCW 7:333).

The object, of course, cannot be treated as a classical system. What can be said about it –

and the significance of the distinction between the way we describe the object and the

apparatus – makes up the fourth corner of support for my interpretation of classical

concepts, to which we now turn.

According to Bohr the very meaning of the term ‘measurement’ requires that we

be able to apply the concepts of classical physics in the context of measurements:

3636

[H]owever far the phenomena transcend the scope of classical physicalexplanation, the account of all evidence must be expressed in classical terms. Theargument is simply that by the word “experiment” we refer to a situation wherewe can tell others what we have done and what we have learned and that,therefore, the account of the experimental arrangement and of the results of theobservations must be expressed in unambiguous language with suitableapplication of the terminology of classical physics. (Bohr 1949, p. 209; BCW7:349)

Classical concepts are required because in their absence we cannot “tell others what we

have done and what we have learned.” The heart of Bohr’s concerns regarding

communicability lies in the issue of interpreting the quantum formalism. At issue is how

we are to invest the abstract symbols of quantum theory with meaning. To highlight the

interpretive significance of Bohr’s account let us first consider a standard modern gloss

on the difficulties we face in interpreting the quantum formalism.

It is now common to view interpretations of quantum mechanics as proposed

solutions to the so-called measurement problem, a typical version of which runs as

follows: Suppose we have a device that measures some property, such as whether a

particle passes through the upper or lower slit in a diaphragm. By definition this means

that if the particle is localized in the upper slit of the diaphragm, then the interaction of

the particle with the apparatus will leave the apparatus in a state indicating this; we

describe this by saying that the pointer needle of the apparatus goes from a “ready” state

to an “up” state. If we assume that the state of the particle is left unchanged, we can

represent this evolution as

|8,|“ready”, Y |8,|“up”,.

7If we invoke a different, nonunitary evolution in measurement interactions – such as vonNeumann’s type I evolution – we then face the problem of specifying in a nonarbitrarymanner what constitutes a measurement.

3737

Likewise, if the particle passes through the lower slit, the pointer needle will evolve into a

“down” state:

|9,|“ready”, Y |9,|“down”,.

The rub lies in the fact that quantum evolution is linear, which implies that if |8, and |9,

are both solutions to the equations of motion, then a linear superposition of these states

such as |8, + |9, (ignoring normalization factors) will also be a solution. However, if the

evolution represented by “Y” is the unitary evolution of quantum theory, then such a

state will evolve as

(|8,+|9,)|“ready”, Y |8,|“up”, + |9,|“down”,.

The state we are left with seems to describe a superposition of “up” and “down” states for

the macroscopic pointer needle; however, we never experience such superposed

macroscopic states. The standard measurement problem, then, is to reconcile the lack of

macroscopic superpositions in our world with a quantum theory that seems to demand

such superpositions.7

By contrast, Bohr’s measurement problem concerns “the possibility of attributing

to each individual measurement result a well-defined meaning in the sense of classical

mechanics” (Bohr and Rosenfeld 1933, p. 359; BCW 7:125). He would claim that what

we have called the standard measurement problem is ill formed from the very start, for

our glib identification of state vectors such as |8, with physical properties such as having

8Bohr himself emphasizes on a number of occasions the primacy of these interpretiveissues. For example, he tells us, “The main purpose of the [Como lecture] is to show that[the] feature of complementarity is essential for the consistent interpretation of thequantum-theoretical methods” (Bohr 1934 p. 11; BCW 6:289, emphasis added).

3838

a location within the upper slit of a diaphragm stands in need of justification.

Mathematical objects in the quantum formalism, such as a wave function or a vector in a

Hilbert space, can be interpreted only with great care; and Bohr’s primary concern is

precisely to provide such an “unambiguous interpretation of the symbols of quantum

mechanics” (1935 p. 701; BCW 7:297).8

The importance of having a legitimate classical mechanical description of our

measuring arrangements is that these descriptions give meaning to an otherwise merely

symbolic formalism; that is, our classical descriptions give us an interpretation of

quantum theory. As Bohr tells us, “the appropriate physical interpretation of the symbolic

quantum-mechanical formalism amounts only to predictions, of determinate or statistical

character, pertaining to individual phenomena appearing under conditions defined by

classical physical concepts” (1949, p. 238; BCW 7:378, emphasis added). We can, for

example, specify the probability of an electron having some given position upon its

impact with a photographic screen, or the probability of the electric field in a region

having a particular value in a particular direction; such statements are meaningful because

we know what positions are (everyday common language) and what an electric field value

is (transfer of momentum in our classical physical theory).

The key to interpreting quantum theory, according to Bohr, is the recognition that

our classical descriptions are still legitimate in certain limited domains, and it is clear that

9We shall return to the question of Bohr’s justification for this claim in the final chapter. At root, however, it is simply based on the observation that classical mechanics fails tooffer an account of transitions between stationary states of atoms, electron interferencepatterns, and other quantum phenomena.

3939

the classical descriptions he has in mind are precisely the laws and properties of our

classical physical theories. He tells us that because “the unambiguous interpretation of

any measurement must be essentially framed in terms of the classical physical theories

. . . we may say that in this sense the language of Newton and Maxwell will remain the

language of physicists for all time” (Bohr 1931, p. 692; BCW 6:360). The dynamical

laws of Newton and Maxwell have only limited applicability, but only by ensuring that

we can apply these laws in the contexts of our measurement – i.e., by ensuring that we are

safely within the classical limit of quantum theory – can we meaningfully apply the

quantum formalism.

1.4: Measurement and Disturbance

A measurement – by definition, according to Bohr – must employ instruments

whose functioning is solidly in this classically describable realm. However, it is clear that

classical descriptions cannot be universally valid.9 Since measurements allow us to

attribute classical values to objects, it is clear that there will have to be some principled

limitation on measurements that mirrors the limited applicability of classical concepts.

On Bohr’s account, this limitation is enforced by the inescapable interaction between the

apparatus and the object under investigation. This interaction implies a disturbance – not,

properly speaking, of the properties of the object – but of the functioning of other

4040

instruments that might be coupled to the object. It is important to clarify the actual role

that disturbance plays in Bohr’s account of the limits of applicability of classical concepts

because I shall argue later in the dissertation that BHC should not be considered an

example of Bohr’s complementarity because BHC offers no comparable mechanism to

ensure the mutual incompatibility of measurements that allow us to attribute

complementary descriptions.

In our discussion of the two-slit experiment we offered a heuristic account of the

incompatibility of measurements of complementary observables by arguing that a

diaphragm cannot be both fixed and free, and the former is required for a position

measurement while the latter is required for a momentum measurement. Section 1.2

offered a more technical example of this incompatibility based on the interaction between

charged test bodies and the electromagnetic field. Let us look at a formal characterization

of a quantum mechanical interaction to see how these limitations of comeasurablity arise

more generally.

Suppose we are interested in measuring the value of an observable, represented by

the operator x$, belonging to the object system. This will require us to couple the

observable we are interested in to some observable, represented by y$, of an apparatus

system – where the value of this apparatus variable is directly observable, or at least can

be measured more directly than the property of the object. Such a coupling is given by

the following interaction Hamiltonian:

H$ I = k x$ p$y,

10It is noteworthy that Heisenberg’s equation (1.3) is isomorphic to Hamilton’s equationof motion. Indeed, Bohr tells us that the classical and quantum equations are actuallyidentical: “In this formalism, the canonical equations of classical mechanics

dpi/dt = ! MH/Mqi , dqi/dt = MH/Mpi

are maintained unaltered, and the quantum of action is only introduced in the so-calledcommutation rules

pi qi ! qi pi = h/2Bi for any pair of canonically conjugate variables” (Bohr 1939, p. 14; BCW 7:306). It thusappears that Bohr views quantum mechanics as retaining the dynamics of classicalmechanics, while the quantum nature of the theory is expressed solely in the limitationswe face in attributing classical values to the properties of the system. We shall return tothis point below.

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where k is a coupling constant and p$y is the operator representing the variable conjugate to

y$. If this coupling is strong enough during the period of interaction we can ignore the free

evolution part of the Hamiltonian, and the evolution of y$ will be given by Heisenberg’s

equation:10

dy$/dt = MH$ I / Mp$y = k x$. (1.3)

Thus the value of y$ will become correlated with the value of x$, and the greater the

strength and duration of the coupling, the stronger the correlation will become. If we can

directly observe (or unambiguously measure) the value of y$ then we can, in principle,

infer the value of x$ to any desired degree of precision.

Let us see what happens if we try to couple apparatus variables to both the object

observable x$ and its conjugate momentum p$x. Such an interaction is given by the

interaction Hamiltonian

H$ IN = k x$ p$y + kN p$x p$z,

where p$z is the conjugate momentum to our new apparatus variable z$. The evolution of

our apparatus variable í will again be given by (1.3), but now the time rate of change of x$

4242

cannot be neglected. This will be given by

dx$/dt = MH$ IN / Mp$x = kN p$z.

Thus the apparatus variable í will not only be coupled to the object variable x$, but it will

also be coupled to the apparatus momentum p$z. Likewise, the apparatus variable z$ will be

coupled to p$y. Our ability to infer the values of x$ and p$x will therefore be limited by our

imprecise knowledge of the apparatus variables – and the validity of the uncertainty

principles for the apparatus will imply that it is impossible to measure conjugate variables

of the object more precisely than allowed by the uncertainty principles.

Notice, however, that nothing prevents us from coupling different instruments to

“incompatible” observables of an object, that is, observables represented by

noncommuting operators. The important point is that if these instruments are to serve

their purpose as measuring devices, then they cannot both couple to incompatible

observables of a system – or else one or both of them will be unable to function properly.

The interaction between the apparatus and the object is “uncontrollable” to the

extent that the quantum uncertainties play a significant role in our description of the

behavior in question. When we have only one apparatus coupled to our object system,

these uncertainties can be neglected once the interaction has established a sufficient

correlation to allow us to neglect the uncertainties in the initial and final values of í. In

the case of our attempt at measuring both x$ and p$x, however, there never comes a point at

which the reciprocal uncertainties of í and p$y, and z$ and p$z, become irrelevant for our

description.

11We know this to be problematic because of the Bell and Kochen-Specker theorems.

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A number of philosophers have claimed that Bohr’s emphasis on the

uncontrollable interaction involved in measurement commits him to a problematic

disturbance view of the origin of quantum uncertainties. Mara Beller, for example, in her

1999 book develops a claim – put forward earlier in Beller and Fine (1994) and Fine

(1986) – that the “concept of disturbance” was central to Bohr’s interpretation of quantum

mechanics until he was forced to abandon this view by the 1935 EPR paper and retreat

into operationalism.

The disturbance view that Beller and Fine have in mind here is the view that the

quantum object has objectively definite values of all its properties (e.g., both of position

and momentum) at all times – or at least when it is not interacting with some other

system. However, we cannot come to know these precise property values – i.e., we

cannot exactly measure (all) of these values – because any measurement of a property will

uncontrollably disturb the values of any complementary property. In Beller’s words,

“The concept of disturbance, inaugurated in Heisenberg’s uncertainty paper, is an ill-fated

and inconsistent one: it presupposes the existence of objective exact values that are

changed by measurement, contrary to the desired conclusion of indeterminacy” (1999, p.

156). The problematic aspect of this interpretation of quantum mechanics is not the claim

that measurements (potentially) disturb the properties of the object system, but rather the

implicit assumption that before and after the “disturbance” the property has some definite,

although unknown, value.11 We therefore need not only to ask whether Bohr believed

that there was a disturbance of these properties in measurement, but also whether – if he

4444

does make this claim – he is then forced to accept the problematic position that all

properties have objectively definite (classical) values before and after measurement.

Let us consider the evidence that Beller provides for the claim that prior to 1935

Bohr believed that quantum systems had objective exact values that will be

uncontrollably disturbed if one measures some complementary observable. Her account

of Bohr’s early view is the following: “Bohr’s early description of the nature of

measurement invoked the realistic imagery of existing phenomena (no operational

definition of concepts yet!) and of disturbance, or finite changes in the phenomena,

during measurement” (Beller 1999, p. 157). Beller supports this reading with a quotation

from an unpublished version of the Como lecture: “the quantum postulate implies that no

observation of atomic phenomena is possible without their essential disturbance” (BCW

6:91, Beller’s italics). She then goes on to claim that this phrasing “is not an accidental,

unhappy choice of terminology – the idea persists in all of Bohr’s writings in the late

1920s to early 1930s” (Beller 1999, p. 157). However, the fact that Bohr’s reference to

an “essential disturbance” was expunged from the published versions of the Como lecture

seems to indicate that Bohr may well have believed that this was an unhappy choice of

terminology. The version published in Nature reads instead, “the quantum postulate

implies that any observation of atomic phenomena will involve an interaction with the

agency of observation not to be neglected” (Bohr 1928, p. 580; BCW, 6: 148). A natural

explanation for Bohr’s elimination of the term “disturbance” from the final draft is

precisely that this term might be taken to imply a preexisting, well-defined value that is

then perturbed – while an “interaction” need not carry this same connotation. However,

12This has been cogently argued by Howard (1979, pp. 188-190).

4545

the most compelling reason for denying that Bohr held this problematic view lies in the

fact that he sees complementarity as a limitation on definability, and not primarily on

measurability.12

From the birth of his account of complementarity, Bohr consistently argued that

quantum theory introduces a fundamental limitation on our descriptions of nature and not

merely a limitation on our ability to measure some actual, but unknown, state of the wold.

The epistemological lesson that Bohr believes is forced on us by the existence of Planck’s

constant is that our classical concepts, e.g., our spatiotemporal descriptions or the

attribution of energy-momentum, are fundamentally limited. Contrary to Beller and

Fine’s claims, this emphasis on the limits of definability is not a shift in position due to

the argument presented by EPR – but is the principal message of Bohr’s account of

complementarity from 1927 on. Not only do we find this point emphasized at the very

beginning of his Como lecture, but it is presented as the central lesson of quantum theory

in all of his papers up to, as well as after, his 1935 response to EPR:

The quantum theory is characterised by the acknowledgment of a fundamentallimitation in the classical physical ideas when applied to atomic phenomena.(Bohr 1928, p. 580; BCW 6:148)

[T]he very recognition of the limited divisibility of physical processes,symbolized by the quantum of action, has justified the old doubt as to the range ofour ordinary forms of perception when applied to atomic phenomena. (1929b, p.93; BCW 6:209)

In quantum mechanics . . . we are concerned with the essential incompatibilitybetween the elementary laws of atomic stability and the use of the classicalmechanical concepts on which all measurements must be interpreted. (1932, p.377; BCW 6:401)

4646

Obviously, these facts not only set a limit to the extent of the informationobtainable by measurements, but they also set a limit to the meaning which wemay attribute to such information. (1934, p. 18; BCW 6:296, Bohr’s emphasis)

Indeed we have in each experimental arrangement suited for the study of properquantum phenomena not merely to do with an ignorance of the value of certainphysical quantities, but with the impossibility of defining these quantities in anunambiguous way. (1935, p. 699; BCW 7:295)

Complementarity was always for Bohr essentially an account of the limitations of

“classical physical ideas,” “forms of perception,” “classical mechanical concepts,” and

the “meaning” and possibility of “defining” physical quantities.

I have argued that the essence of the indivisible nature of the interaction between

the object and the apparatus lies in the unavoidable disturbance of one measuring

arrangement by another, where this “disturbance” is mediated by the object system that

both measuring apparatuses are coupled to. I have also argued, however, that Bohr’s

thesis is about the limits of concepts and not merely of measurability. How then are we

to ground these claims regarding the limitations of our classical descriptions, and what is

the relevance of the fundamental incompatibility of certain measurements? A common

misreading of Bohr has it that the limits of measurability imply the limits of meaningful

application of concepts. It is to this misreading and my suggested alternative that we now

turn.

1.5: Uncontrollable Interactions and the Limits of Classical Concepts

We conclude this chapter on Bohr’s complementarity by addressing the question

of whether his position amounts to some form of verificationism or operationalism. This

13For example, a concept of length would be essentially tied to the procedure of layingmeasuring rods end to end between two points. The foundational account ofoperationalism is Bridgman (1927).

4747

topic is important for two reasons: First, Bohr has been charged by some historians and

philosophers of science with adopting an ill-supported version of positivism. However,

an understanding of Bohr’s justifications for his positions reveals that these charges miss

the mark. Second, several advocates of BHC appear to appeal to some form of a

verification principle, arguing (explicitly or implicitly) that values that are impossible to

measure are meaningless, or not physically real; and they seem to assume that these

arguments are consistent with Bohr’s philosophy. Although Bohr does occasionally

appeal to some form of verificationism, such positivist principles play no role in his

considered account of complementarity.

Operationalism is the claim that our concepts of certain properties get their

meaning from a specification of the procedures, or operations, that one would have to

perform to ascertain the value of the property in question.13 Verificationism is the closely

related claim that statements that are in principle unverifiable are meaningless. It is not

difficult to see why Bohr is often read as a verificationist. Consider, for example, the

following passage from his response to EPR:

But even at this stage there is essentially the question of an influence on the veryconditions which define the possible types of predictions regarding the futurebehavior of the system. . . . [T]hese conditions constitute an inherent element ofthe description of any phenomenon to which the term “physical reality” can beproperly attached. (1935, p. 700; BCW 7:296, Bohr’s emphasis)

His emphasized reference to the “possible . . . predictions” of the behavior of the system

suggests an appeal to a verificationism – and the close tie between measurability and

14Such a charge has been leveled by both Beller and Fine. For example, Fine argues thatBohr’s account here, “is virtually textbook neopositivism. For Bohr simply identifies theattribution of properties with the possible types of predictions of future behavior” (Fine1986, p. 34).

15Note that this passage was written two years before the EPR paper. Thus even if onedoes believe (erroneously, in my view) that Bohr was committed to operationalism, itwould seem to be a (further) mistake to follow Fine and Beller in believing that this“shift” was primarily a response to EPR.

4848

definability invites a charge of operationalism.14 Indeed, one can find even more

problematic examples of apparent verificationist moves. At one point in the argument

regarding the measurability of quantum fields, he and Rosenfeld reach an inequality for

the uncertainty of the measured field values that, while smaller than the limit suggested

by Landau and Peierls, is still finite. Bohr and Rosenfeld claim that if this inequality

were to be considered as an unavoidable limit on the accuracy of measurement,we should still arrive at the conclusion, in agreement with the view of theseauthors, that the quantum-electromagnetic formalism admits of no test in theproperly quantum domain, and that therefore physical reality can be ascribed tothe entire field theory only in the classical limit. (1933, p. 386; BCW 7:152)

They go on to demonstrate that one can do better than this limit by installing an

appropriately calibrated spring, but this nevertheless is an indication that Bohr sometimes

slips into equating “physical reality” with measurability or predictability.15

But in spite of these passages, Bohr’s more considered view is that the

fundamental limitation is one of classical concepts and not merely of measurability. In

his response to EPR, for example, Bohr quickly abandons any reference to predictability,

and instead speaks of “possibilities of unambiguous interpretation of measurements”

(Bohr 1935, p. 700; BCW 7:296, emphasis added). Elsewhere Bohr is more explicit in

claiming that the limitations of measurement are a reflection, or a “consequence,” of the

4949

limits of our concepts, and not the other way around: “It is, therefore, an inevitable

consequence of the limited applicability of the classical concepts that the results

attainable by any measurement of atomic quantities are subject to an inherent limitation”

(1929, p. 95; BCW 6:211).

Bohr sees these conceptual limits being grounded in the finite value of S and the

existence of phenomena that are unexplainable by classical physics. In the case of

relativity theory, the breakdown of our intuitive distinction between space and time can

be traced to the existence of a maximum velocity for all physical processes, a velocity

that is the same for all inertial observers. In a similar way, Bohr believes that the

breakdown of our physical descriptions in terms of the conjugate variables of classical

mechanics can be traced to the existence of a minimum value of action. That is, the

quantum nature of the world is symbolized by Planck’s constant, which “is only

introduced in the so-called commutation rules . . . for any pair of canonically conjugate

variables” (Bohr 1939, p. 14; BCW 7:306, emphasis added). The commutation relations,

and the uncertainty principles that are derived from them, express the limits of our ability

to apply classical concepts.

With the failure of our classical deterministic description comes a necessary

appeal to probabilistic reasoning. The “essentially statistical nature” of quantum theory,

according to Bohr, is “a direct consequence of the fact that the commutation rules prevent

us to identify at any instant more than half of the symbols representing the canonical

variables with definite values of the corresponding classical values” (1939, p. 15; BCW

7:307). The picture here is that the deterministic character of our classical theories rests

16A rigorous account of this would require a number of qualifications and would enter usinto debates that go beyond our concerns here. The interested reader is referred toEarman (1986) and Frisch (2003).

5050

on the fact that a specification of the initial values of the generalized coordinates and their

conjugate momenta, together with the Hamiltonian of the system, fixes the future

evolution of the system.16 The limitations facing these concepts of coordinates and

momenta then imply a limitation on the facts that would (in a classical world) fix the

future state of the system. Therefore, a theory that captures these limitations – viz.,

quantum mechanics – will have to be a statistical theory.

We can set aside for the moment the question of how compelling this picture of

Bohr’s is and focus instead on the issue of how this is related to his account of

disturbance in measurement. Recall that in his characterization of the measurement

problem in quantum mechanics Bohr tells us that “the quantum-imposed interaction,

uncontrollable in principle, between instrument and object is fully taken into account

through the influence of each measuring process on the statistical expectations testable in

succeeding measurements” (Bohr and Rosenfeld 1933, p. 359; BCW 7:125, emphasis

added). The interaction cannot be described by classical physics (and therefore we cannot

compensate for it), but we can take account of it nonetheless, precisely by specifying the

potential influence of our measurement on future measurement outcomes. So, for

example, the displacement DxI of our charged test body is not to be interpreted as an

actual (though unknown) position of the body, but is instead a partial measure of the

influence that this body can have on other test bodies arranged to reveal average field

values. Likewise, we are not to think of the so-called uncertainties of our apparatus

5151

variables í and p$y (and z$ and p$z) as mere measures of our ignorance: instead we are to see

them as a characterization of the limitations that our classical concepts face due to the

nonclassical atomistic nature of quantum interactions – limitations that manifest

themselves when we try to use these interactions to attribute values to x$ and p$x, for

example.

The importance, for Bohr, of the limitations imposed by this uncontrollable

interaction on measurement lies in the fact that in the absence of such limitations the

world would have to be governed by classical mechanics. Note that in his response to

EPR he is concerned to show the limits of

unambiguous interpretation of measurements, compatible with the finite anduncontrollable interaction between the objects and the measuring instruments inthe field of quantum theory. In fact, it is only the mutual exclusion of any twoexperimental procedures, permitting the unambiguous definition ofcomplementary physical quantities, which provides room for new physical laws. (Bohr 1935, p. 700; BCW 7:296, emphasis added)

Because measurements allow us to attribute classical values to systems, if there were no

limits on the possibilities of measurement, then it would be possible, in principle, to offer

a completely classical (and deterministic) description of the system. If we are to make

room for nonclassical phenomena – as we must, given the experimental evidence – then

we will have to establish fundamental limitations on the possibilities of measurement.

These limitations are to be found in the uncontrollable interactions symbolized by the

uncertainty relations.

To return to operationalism, then, we can see that, despite some potentially

misleading similarities with Bohr’s account, the operationalist program fails to capture

5252

the essence of complementarity. What sets Bohr’s account apart from this view is that

the applicability of classical concepts does not rest – primarily, at least – on our ability to

perform certain operations, but rather on the demonstrable applicability of the laws of

classical mechanics in certain situations. If classical mechanics were universally

applicable – that is, if there were no finite constant of action – then there would be no

fundamental need to investigate the possibilities of measurement to establish the domain

of applicability of our concepts or their meanings.

Verificationism also fails to capture the substance or grounding of Bohr’s

complementarity: He does not argue that classical concepts have limited applicability

because there are limits to the possibilities of their verification. His arguments are

intended to show that one cannot measure property values more accurately than the

uncertainty relations allow; however, he does not argue that these limits on measurability

directly imply that claims about such properties are meaningless. The argument for the

limits of the applicability of classical property values rests on the experimental

verification of quantum phenomena that cannot be accounted for classically.

Nonclassical behavior occurs in the real world; therefore classical mechanics cannot be

universally valid. It is not a question of lack of verifiability implying meaninglessness,

but rather the falsity of classical predictions that implies limits to the accuracy, or

applicability, of classical concepts.

Thus, the crucial points that a legitimate appeal to Bohr’s complementarity will

have to include are, first, an account of the classical laws and properties that can be

recovered in an appropriate limit of the quantum theory, and second, an account of how

5353

the uncontrollable nature of quantum interactions – symbolized by the noncommutativity

of the appropriate operators – prevents the application of these classical concepts in

measurements. Before we can evaluate BHC on these counts, however, we must lay the

groundwork for an explication and evaluation of these proposals. The first preparatory

step in this direction focuses on quantum field theory, which is the topic of Chapter 2.

54

CHAPTER 2

QUANTUM FIELD THEORY

The primary goal of this chapter is to offer a brief overview of QFT (quantum

field theory) to lay the groundwork for topics that will be addressed later in the

dissertation. More specifically, our discussion of Hawking radiation in Chapter 3 and the

argument for information loss in black holes in Chapter 4 will require some vocabulary

and basic concepts from QFT, as will our discussion of black hole complementarity. This

chapter is not intended to be comprehensive introduction to the theory, however. A more

complete and rigorous account can be found in any introductory quantum field theory

textbook, e.g., Peskin and Schroeder (1995) or Ryder (1985). A nice introduction to QFT

with an emphasis on interpretive issues can be found in Teller (1995).

The structure of this chapter is as follows: We begin by stepping through the

quantization of a free field and the Fock space representation, which facilitates an

interpretation of the quantum field in terms of particles. Section 2.2 briefly treats field

interactions and sketches the nature and significance of the S-matrix in QFT. These

points will be important for understanding black hole complementarity, as the existence

of an S-matrix description of black hole formation and evaporation is a central tenet of

these proposals. Section 2.3 addresses problems with formulating QFT using only the

55

apparatus of operators in a Hilbert space, and briefly discusses the more general algebraic

version of the theory. However, for reasons that will be discussed, we shall have little

need to appeal to this more general formulation in the rest of the dissertation. In the final

section we extract a few conceptual points that will be of use later: we highlight the

significance of microcausality, which plays an important role in driving the argument for

information loss in black holes, and we discuss the extension of complementarity into the

quantum theory of fields.

2.1: Free QFT and Fock Space

Quantum field theory is one of our most fundamental and most precisely verified

physical theories. Quantum electrodynamics, for example, – which describes electrons

and the electromagnetic field, as well as the interactions between the two – yields

predictions that accord with experiment to an accuracy of one part in a hundred million.

For the purposes of our discussion, however, we need not consider the details of realistic

theories such as this, but can content ourselves with the much simpler example of a free

scalar field, n(x), governed by the Lagrangian

� = ½ (M:n)2 - ½ m2 n2.

From this Lagrangian we derive an equation of motion, given by

(M:M: + m2) n = 0,

which is known as the Klein-Gordon equation. The solutions, n(x), of this wave equation

will be assignments of a numerical value to each point of spacetime x/(t, x).

56

To quantize this field we need to convert these solutions into Hilbert space

operators, N(x), that obey the following equal-time canonical commutation relations with

their conjugate momenta, B(x) / = dN(x)/dt:

[N(t, x), B(t, xN)] = i*3(x!xN);

[N(t, x), N(t, xN)] = [B(t, x), B(t, xN)] = 0.(2.1)

These commutation relations define the algebraic foundation of our quantum theory, but

they do not uniquely determine a single theory. (We shall pick up this point below in

Section 2.3.) To generate our quantum Hilbert space, we need to decompose the field

operators in terms of a complete set of solutions to the wave equation. In the Minkowski

spacetime under consideration we can use an inertial observer’s coordinates to find the

positive-frequency modes of the wave equation:

(2.2)

where k< x< = k0x0 ! kAx, and Tk = (kAk + m2)1/2 = k0. These plane-wave solutions allow

the Fourier decomposition of the field operators:

N(x) = Id3k (âk ek + âk† ek*). (2.3)

The Fourier coefficients âk† and âk are operators that allow us to generate a basis of our

Hilbert space – and open the door to a particle interpretation of quantum fields.

When confronting a standard harmonic oscillator problem, one can generate a

solution by introducing “raising” and “lowering” operators that represent the addition or

subtraction of an excited mode of the oscillator. In the same way, one can view âk† and âk

as creation and annihilation operators that represent the generation or destruction

17Note that talk of “particles” in QFT is somewhat loose for reasons that will be discussedbelow in Section 2.3

57

(2.4)

(respectively) a single particle17 of momentum (or wave number) k. This allows us to

represent states of any (finite) number of particles. We begin by specifying the vacuum

state |0,, which is annihilated by any annihilation operator: âk|0, = 0, for any âk. (We can

view this as capturing the fact that in the vacuum state there are no particles that might be

removed.) We then generate particle states of definite momentum by applying the

appropriate creation operators to the vacuum state, |k, = âk†|0,. The set of all possible

combinations of creation operators applied to the vacuum provides us with a basis for our

Hilbert space. The states generated in this way, containing a definite number of particles

all with definite momenta, are referred to as Fock states (and the Hilbert space thus

generated is Fock space).

To bolster our interpretation of Fock states in terms of particles we can introduce

a “number operator” Nk = âk†âk; Fock states will be eigenvectors of this operator with an

eigenvalue that corresponds to the number of particles with momentum k contained in

that state. This can be seen by rewriting the Klein-Gordon Hamiltonian,

H = ½Id3x (B2 + (LN)2 + m2N2 ), in terms of our creation and annihilation operators:

Although the ½ term on the right causes this integral to diverge, one typically argues that

this infinite zero-point energy can be ignored because only energy differences are

measurable; this move is equivalent to imposing “normal ordering,” which requires us to

58

write all annihilation operators to the right of all creation operators. Without the ½ term

the Hamiltonian (2.4) is simply the (continuous) infinite sum over the number operator,

which specifies the number of particles with momentum k, multiplied by the energy

associated with that momentum. Thus our particle interpretation allows us to say that the

Hamiltonian is the straightforward sum of the energies of all particles present, as it should

be.

It is important to recognize the significance of our ability to decompose the field

into positive and negative frequency parts as in Equation (2.3). This allows us to define

our free field vacuum and our creation and annihilation operators, all of which continue

to play an important role when we move to interacting theories. However, this

decomposition assumes the existence of a global time coordinate to distinguish the

positive and negative frequencies. Although this is unproblematic in flat spacetime

(because we can appeal to Poincaré invariance to privilege inertial observers), in curved

spacetime such privileged decompositions will not generally exist. Indeed, the derivation

of Hawking radiation (discussed below in Section 3.4) involves essentially nothing more

than rewriting the vacuum of Minkowski spacetime in a decomposition appropriate for a

distant observer far outside a black hole at late times. Many of the conceptual issues

surrounding QFT in curved spacetime (e.g., the ontological status of particles and the

information loss paradox) are intricately tied to the question of how we are to view

different decompositions of the quantum field. A further worry, to be addressed below in

Section 2.3, is based on the fact that different decompositions typically result in different

Hilbert spaces that are unitarily inequivalent.

59

2.2: Interacting QFT and the S-matrix

In the forgoing section we implicitly made use of the Heisenberg picture of time

evolution in which the time dependence is carried by the operators representing the

observables of the system. Thus our field operators N(x) are functions of time and are

assumed to evolve unitarily under the Hamiltonian of the system:

N(x) = N(x, t) = U !1 N(x, t0)U = ei  tN(x, t0)e!i  t.

This is to be contrasted with the Schrödinger picture, in which the observables are time

independent and the state undergoes an evolution represented by the unitary

transformation of the state vector, |R(t), = U |R(t0), – or, more generally, as the unitary

transformation of a density matrix, D(t) = UD(t0)U !1. Because the measurable quantities

in quantum theory are expectation values – given by +R|Â|R, or Tr(DÂ), where  is an

observable – these two pictures are experimentally equivalent.

A third picture of time evolution is available when we consider a system that

includes interactions. If our full Hamiltonian consists of a free part and an interaction

part,

H$ = H$ 0 + H$ I,

we can employ the interaction picture in which the operators evolve under free evolution

(governed by H$ 0) and the states’ evolution is given by the interaction Hamiltonian, H$ I .

Let us begin with some number of particles (typically two) that are sufficiently

separated so that they can be treated as free particles. We bring them together, they

interact, and then we consider the results of the interaction at late times – i.e., we wait

until the particles are again sufficiently separated to consider them free. The transition

60

from the (effectively) free in-states, to the (effectively) free out-states is represented by a

unitary scattering matrix, or S-matrix. Its elements are the transition amplitudes from

Fock states representing a definite number of particles with definite momenta in the

distant past to states similarly representing definite numbers of (free) particles in the

distant future. Thus, if nk1 is the number of particles with momentum k1, then a generic

Fock state in our Hilbert space representing particles in the distant past will be given by

|nk1, nk2, . . . ,. If we offer a similar specification of the basis vectors of our final Fock

space, |nk1N, nk2N, . . . ,, then a generic element of the S-matrix will be given by

+nk1N, nk2N, . . . | nk1, nk2, . . . ,.

Filling in the transition amplitudes for all Fock states in our early time Fock space and all

Fock states in our final space amounts to specifying the operator S that takes us from any

asymptotic in-state, |in,, to any asymptotic out-state, |out,:

|out, = S |in, or Dout = S Din S !1. (2.5)

The significance of the S-matrix lies in the fact that arguably all our empirical support for

QFT lies in confirmations of scattering amplitudes.

An important assumption (or axiom, depending on how it is presented) in this

standard picture of the S-matrix is that the set of free asymptotic in-states and the set of

asymptotic out-states each form a basis for the Hilbert space of the full interacting

system. This assumption, referred to as asymptotic completeness, is equivalent to

assuming that measurements performed in one of the asymptotic regions (typically it is

the asymptotically late-time region that we are concerned with) can, in principle, specify

the complete state of the field. Of course, one would need to perform repeated

61

measurements on an ensemble of identically prepared systems to establish the frequency

of each outcome, and one would also need to establish the relative phases by performing

measurements incompatible with the first on some members of the ensemble. But were

we to perform these measurements we should discover the original quantum state of the

system, and in this sense all the information concerning the state of the field has been

retained. As we shall see in Chapter 4, Hawking suggests that the information loss

paradox implies that a theory of quantum gravity describing evaporating black holes will

have to give up the axiom of asymptotic completeness. This is intended to correspond to

the expectation that a complete set of late-time measurements (even on an ensemble of

identical systems) will not allow us to infer the original state of the system

Once we include interactions in our field theory, we are beset by infinities that

pose both technical and conceptual problems. The technical difficulties are tamed, when

possible, through a process known as renormalization. While we cannot enter into the

details and difficulties of this issue here (the interested reader is referred to Teller 1995

and references therein), the general strategy can be quickly sketched. Suppose we have a

charged point particle such as an electron, and we want to calculate its effective mass.

Because the particle interacts with the electromagnetic field, we must consider the energy

of this interaction in addition to the “bare” mass of the electron. When we attempt to

calculate the energy of this interaction we find that the energy diverges. (Recall that this

fact forced Bohr and Rosenfeld to eschew point particles and restrict themselves to

finitely extended test bodies.) We proceed by noting that the bare mass of the electron

can never be measured, for it will always be embedded in an electromagnetic field. This

62

allows us (in a manner of speaking) to hide the troublesome infinities in the unobservable

bare mass of the electron. Once we measure the effective mass of an electron, we can

(loosely speaking) calculate the difference between that finite value and the infinite value

that the theory predicts if the electron is given a finite value, and subtract this difference

from all of our future calculations. Surprising though this may seem (and it can be made

to sound less surprising – see, e.g., Huggett 1996), the procedure is astonishingly

successful.

2.3: Unitary Equivalence and Algebraic QFT

The structure of QFT is largely given by the equal-time commutation relations

(2.1). However these relations are not sufficient to pick out a single representation of the

commutation relations, where a “representation” is given by a Hilbert space , and a set

of (bounded) operators N(x), B(x) satisfying these relations.

We are faced with a parallel situation in particle quantum mechanics in that the

canonical commutation relations,

[pi, qj] = -iS*ij; [pi, pj] = [qi, qj] = 0,

do not specify a single Hilbert space. However when our system has only a finite number

of degrees of freedom (e.g., when we are considering a finite number of particles) we

have a theorem – the Stone-von Neumann theorem – that guarantees that all Hilbert space

representations of the commutation relations will be unitarily equivalent. This means that

for any two representations there will be some unitary operator that maps the states and

observables of the first representation onto the states and observables of the second; this

63

guarantees that the empirical predictions based on these two representations will be the

same.

Once we move to a field theory we have an infinite number of degrees of freedom,

and the Stone-von Neumann uniqueness theorem no longer holds. In this case there will

generally be many unitarily inequivalent representations of the canonical commutation

relations. These different Hilbert space formulations will generally assign different

probabilities to the outcomes of measurements – because there is no unitary mapping

from the states and operators of the one representation to those of the other – which raises

worries about how we are to find the “correct” representation of our theory.

One response to these worries is to reduce our reliance on a Hilbert space

formulation of the theory, and instead focus on the algebra of observables of the theory.

Such a program has been developed by Haag (1992), for example, and is referred to as

algebraic QFT. Haag’s approach takes into account the Bohr and Rosenfeld concern

about the unobservability of field values at a point by taking the observables to be field

values “smeared out” by well-behaved test functions that vanish outside of some region O.

The algebra of observables A is then defined by the equal-time commutation relations and

the requirement that if O1 d O2, then A (O1) dA (O2). One can define states as a maps from

the algebra of observables to real numbers, the expectation values of the observables for

that state of the system. The hope is that this more general framework will allow us to

overcome problems facing the more limited Hilbert space approach.

One such problem arises when we consider QFT in curved spacetimes. In such

spacetimes we do not have an invariant way of decomposing our field into positive and

64

negative frequency parts, as in Equation (2.3). This means that we do not have a

preferred way of defining our creation and annihilation operators or our vacuum. This in

turn means that we have no way of picking out a single Fock space, and thus there seems

to be no physically significant particle interpretation of such a theory. This by itself

might not trouble us if we are prepared to accept that QFT is fundamentally a theory of

fields and not of particles, but we face a further worry in that Hilbert spaces defined on

different spacelike hyperplanes can be unitarily inequivalent. Thus it would appear that

the evolution of the state of the field from one time (i.e., from one spacelike hyperplane)

to another cannot be represented as the evolution of a quantum state in a (single) Hilbert

space. The fact that evolutions of this type can be treated in algebraic QFT gives us

reason to believe that a move to this more general framework might be necessary if we

wish to incorporate gravity into our quantum theory.

Nonetheless, the significance of the algebraic formulation can be challenged on a

number of points, three of which are particularly relevant for the project we have before

us. First, whenever one wishes to perform actual calculations one must adopt a Hilbert

space formulation; that is, despite its promise, Algebraic QFT has not yet delivered a

formalism that allows us to compare it usefully with the real world. While the algebraic

approach may succeed in areas where the Hilbert space formulation fails, and while it has

the advantage of allowing us to prove certain theorems with rigor, it has not yet produced

solutions to any difficulties facing the standard formulation. In the context of this

dissertation, the most significant point is that the vast majority of the proposals we shall

18There is some controversy over the extent to which this claim is true. Ruetsche (2002)follows Summers (2001) in pointing out that even if we find some state TN in ,N thatoffers the same expectation values of some set of observables {Âi} as our original state Tin ,, there is no guarantee that TN and T will continue to be equivalent when we considersome further observables that we might measure in the future. However, this does notseem to impugn the operational indistinguishability of the two representations, for therewill be some other state in ,N, call it T) N, that will be equivalent to T for this enlarged setof observables. The point is that if our predicted expectation values fail to match ourexperimental results we can never know whether we have chosen the wrongrepresentation or merely the wrong state in the correct representation.

65

consider concern a Hilbert space formulation of QFT. Even Hawking’s proposed non-

unitary evolution, discussed below in Chapter 4, is formulated in a Hilbert space.

Second, a theorem of algebraic QFT – Fell’s theorem – tells us that no finite

number of finitely accurate measurements will ever allow us to decide whether we have

chosen the “correct” representation of our algebra. Even though there will be no state in

the unitarily inequivalent Hilbert space that reproduces exactly the expectation values of

our original state, for any arbitrarily small (but finite) number , there will be some state in

the non-equivalent space that reproduces all the expectation values of the actual state to

within ,. Thus the various inequivalent representations will, as a matter of fact, be

experimentally indistinguishable.18

The relevance of this point might be questionable to the extent that we are

interested in offering a rigorous, principled interpretation of the theory. However, we

know that quantum field theory in curved spacetime is only an effective theory that is not

a candidate for a literal description of our world: in our world matter influences the

metric of spacetime, but the curvature of spacetime in these models is independent of the

state of the matter fields. Given that we are only interested in the theory as an

66

approximate account of our world we might argue that a Hilbert space account that has a

precision limited by , (where , can be as small as we like) is just as good as an algebraic

account with unlimited precision: a precise description of an approximate truth may be no

better than a fuzzy description of this same approximation.

This is related to our third reason for questioning the relevance of algebraic QFT

in the context of our project here, namely that it relies on a classical spacetime

background that may be discarded by the full theory of quantum gravity that we are

interested in. Our response to the question of whether we should expect the underlying

full theory of quantum gravity to be unitary is likely to depend on which description of

the world we consider to be more fundamental: the spacetime description offered by GR,

or the Hilbert space description offered by QT. If we are more impressed by the

dynamical spacetime of GR then we are likely to point out that nonunitary evolution

seems to be a fundamental feature of evolution through curved spacetime. We will then

tend to question the significance of unitary evolution and will argue that we should be

looking for ways to run physics without unitary evolution.

The alternative is to view the QT as being more fundamental. One might argue

that quantum mechanics has already forced us to modify our view of all interactions to

which it has been applied – there is no reason to think that a quantum theory of gravity

will retain all the fine details of spacetime that the argument for the nonunitary

equivalence of representations of algebras implies. That is, the worries about

nonunitarity arise because there is no preferred way to pick out positive energy solutions

in curved spacetimes, and indeed there is typically no consistent way to pick positive

67

energy solutions for two different time slices E1 and E2. But if this “inconsistency” is

fine enough (and Fell's theorem seems to indicate that it is “fine enough”) then we might

expect quantum mechanics to introduce enough “course graining” to make everything fit

together smoothly. One of the lessons of quantum mechanics is that the world cannot be

as many ways as it could classically: States are no longer points in phase space, but

instead have a finite phase-space volume. Once the metric tensor of spacetime is

quantized we may no longer be able to specify time slices E1 and E2 finely enough to

establish that the representations of the matter field operators on these slices are not

unitarily equivalent.

The success of unitary quantum theory might suggest that we should expect

quantum gravity to be a truly quantum-mechanical theory in the sense that it is

representable on a Hilbert space with a unitary operator describing the evolution of the

system. One might also be motivated by the hope that one has found a strong candidate

for a full theory of quantum gravity, e.g., string theory, and this theory is manifestly

unitary. A person with this sort of vision of quantum gravity is likely to find arguments

based on effective theories such as QFT in curved spacetimes to be unconvincing, as she

would not expect the full theory of quantum gravity to share the features that these

arguments rest on.

We shall see below that many of the responses to Hawking’s argument for

information loss in black holes follow this line of reasoning – which then focuses the

debate on the question of just how accurate the QFT description is. One of the central

68

issues here will be the validity of the quantum-theoretic ban on superluminal signaling

signified by microcausality.

2.4: Microcausality and Complementarity in QFT

We saw in Section 1.4 that when two observables are represented by operators

that do not commute there will be an “uncontrollable interaction” between the two

measuring arrangements (via the object being measured) that will prevent us from being

able to interpret at least one of these procedures as a measurement of the desired value.

One implication of this is that it is generally possible to signal using measurements of

observables represented by noncommuting operators. A quick and dirty (“dirty” because

it appeals to measurement collapse) argument for this conclusion runs as follows:

Suppose we have two observables Â(x) and B$ (y) that do not commute with each

other. We arrange to have the initial state of our system (e.g., our field) be an eigenstate

of B$ (y) but not of Â(x), and agree that Alice, our observer at event x, will perform the

Â(x) measurement if and only if she wishes to send a positive response to Bob at y. If she

does so then the state of the system will collapse into an eigenstate of Â(x), which will

imply that there will now be a dispersion in the result of the measurement of B$ (y). Of

course, a single measurement may not allow Bob to infer whether or not Alice has

performed her measurement (although if her answer is positive, and he is lucky, a single

measurement will suffice), but they could (in principle) arrange an ensemble of such

measurements that would allow Bob to decide with any degree of confidence desired

whether or not Alice has sent her signal.

69

This is unproblematic as long as x and y are timelike or lightlike related, but if

they are spacelike related this seems to allow superluminal signaling and open the door to

causal paradoxes. Therefore it is generally taken as an axiom of QFT that all spacelike

related observables commute, a condition referred to as microcausality. Note that this is

a condition on the observables of the theory and not (directly) on the field operators

themselves. This distinction is significant for Fermion fields, whose operators

anticommute with each other at spacelike distances. However, because all Fermion field

observables are even functions of the field operators, the observables themselves obey

microcausality. For Boson fields, on the other hand, this distinction is moot, as the field

operators themselves commute when they are spacelike related, and the observables of

the system are arbitrary functions of the field operators.

Microcausality plays an important role in the debate surrounding Hawking’s

information loss paradox. Most participants in the debate are willing to allow the

possibility that at some point this condition might be violated by a full theory of quantum

gravity; however, it is generally accepted that when and where it is possible to offer a

classical description of a spacetime microcausality should hold. We shall return to this

discussion in Chapter 4.

We have characterized observables in QFT in terms of functions (even functions,

in the case of Fermions) of field observables. It might be helpful, however, to say a few

words regarding the sort of measurement procedures with which we are typically

19Along these lines, we might hope for a characterization of what a complete set ofobservables would be for a QFT– i.e., how, in principle, could we specify the completestate of the field? This question is a difficult one because we are restricted to measuringfield values averaged over some region (and thus can never specify N(x) for all points onsome spacelike surface), and a complete measurement in the Fock basis wouldpresumably require a continuously infinite array of particle detectors, the possibility ofwhich is not obvious. We shall therefore set this question aside for the purposes of thisproject.

20An interesting question is whether this will still hold true when we consider QFTs thathave no classical counterpart, by which we mean a Hamiltonian theory over a classicalphase space. This question and the question of how this characterization of a classicaltheory fits with the classical limit described by so-called tree diagrams (i.e., Feynmandiagrams without internal loops), though both important, cannot be pursued further here. A brief answer, which will be made clearer in Chapter 5, is that one can hope to use themore general formulation of complementarity in terms of joint necessity and mutualincompatibility even when there is no classical theory to appeal to. While Bohr neverseems to have explicitly considered such a possibility, modern theories such as QCDpresumably would require a broader account of complementarity if the view is not to berejected.

70

concerned.19 We have already seen in Section 1.2 that the average value of fields such as

the electromagnetic field can be measured in accord with their classical definition. Thus,

for example, one measures the average value of the electric field in some direction by

measuring the momentum imparted to a charged test body during the period of interest,

while compensating for any field effects due to the test body itself. We also saw in that

section that the accuracy of measurements of two average field values is limited by the

magnitude of the commutator between the operators representing those averages. Thus

QFT provides a paradigmatic case of complementary physical descriptions.20

Despite the importance of Bohr and Rosenfeld’s account of the measurability of

quantum fields, no test of quantum theory has ever employed such a measurement

procedure. Our experimental evidence, as mentioned earlier, comes almost exclusively

71

from scattering experiments: we measure the frequency of transitions from incoming

pairs of particles to outgoing collections of particles. Recall that talk of particles in QFT

is to be cashed out in terms of Fock states, which count up the number particles present.

The number operator, Nk = âk†âk, will therefore be the primary observable of interest in

representing scattering experiments.

Because a Fock state is a specification of a number of particles possessing a

certain momentum, we should not be surprised to find that a definite Fock description is

incompatible with a description in terms of definite field values, which are local

quantities in that they depend explicitly on x. Likewise, the operators representing field

values averaged over some spatiotemporal region will fail to commute with operators

representing measurements of the particle content of the field. Thus, if we have a definite

number particles, we will not be able to specify a definite position for those particles, and

vice versa.

This concludes our overview of quantum field theory. We now turn to a

discussion of black hole thermodynamics.

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CHAPTER 3

BLACK HOLE THERMODYNAMICS

Black holes are of foundational and conceptual interest for a number of reasons.

The feature that will be of primary interest for our discussions in this dissertation is the

fact that the interior of a black hole is by definition causally disconnected from the future

evolution of the exterior spacetime. As we shall see in Chapter 4, this fact drives the

argument for information loss. However, black holes are also of interest because, as

purely gravitational entities, they promise to provide a fruitful testing ground for quantum

theories of gravity. Further, there are several issues surrounding the existence of black

holes that indicate that they might offer fundamental insights into the behavior of

quantum gravity. One of these issues is the information loss paradox itself, which

suggests a tension between our current understandings of quantum theory and of general

relativity. A second issue is the existence of a remarkable parallel between laws

describing black holes and laws describing a thermodynamic system. This suggests to

many that the evolution of black holes captures something essential about the quantum

nature of spacetime.

The aim of this chapter is to explicate both these issues and the features of black

holes that will be essential for our later discussions of the information loss paradox and

73

black hole complementarity. We begin in Section 3.1 by discussing various ways that

black holes can be characterized. In particular, we shall consider the so-called membrane

paradigm of black holes in which all of the hole’s properties are ascribed to a fictitious

membrane a small distance above the event horizon. This significance of this picture for

our purposes lies in the fact that it is invoked by black hole complementarians, who want

to invest the membrane with a more robust ontological status. Section 3.2 begins our

discussion of black hole thermodynamics by laying out the laws of black hole evolution

that parallel the case of thermodynamics.

A feature of this evolution that is of fundamental importance both for the

thermodynamic analogy and for the information loss paradox is the fact that black holes

are expected to give off energy in the form of heat. An overview of the derivation of this

effect, referred to as Hawking radiation, requires us to consider a similar effect, the

Unruh effect, that applies to accelerating observers in flat spacetime. Section 3.3

discusses how such accelerations are represented and how they are analogous to the case

of black holes. Section 3.4 and 3.5 then offer an overview of the Unruh effect and the

Hawking effect, respectively.

We conclude in Section 3.6 by returning to black hole thermodynamics and,

specifically, the question of the robustness of the parallel between the second law of black

hole mechanics and the second law of thermodynamics. We shall consider the claim that

independently each of these laws can be violated, but there is a generalized version of the

second law that appears to hold in all physically possible transitions. This claim is

significant not only because it adds support to the claim that black hole thermodynamics

1 For more details on the basics of general relativity and black holes, and for derivationsof the equations below, see, for example, Wald (1984).

74

captures essential features of a full theory of quantum gravity, but also because the

validity of such a law lends support to the possibility that the evaporation of black holes

is a unitary process.

3.1: Holes and Membranes

A naive characterization of a black hole is that of an object of sufficient size and

density to prevent light from escaping its gravitational attraction. This idea goes back at

least to Michell (1784) and Laplace (1796), who argued that the escape velocity of

sufficiently massive bodies could exceed the speed of light. However, it is only after

Einstein developed general relativity that black holes became objects of serious inquiry.

A distinctive feature of black holes from a relativistic point of view is the fact that once

the matter of a collapsing object passes within a critical radius (given by r = 2M for a

Schwarzschild black hole), no physical force can prevent the matter from continuing to

collapse into a point-like singularity. This dramatic behavior was considered physically

untenable by early researchers, and it was only after many years and much hard work,

chiefly by Chandrasekhar, that the prospect of an object undergoing complete

gravitational collapse was taken seriously.

A spacetime is given by a manifold and metric (M, g:<). The vacuum field

equations of general relativity follow from the Lagrangian density1

�E = (!g)1/2 R,

2 We can also consider generalizations of the Einsteinian gravity. For example, dilatongravity is significant both because it arises as the low-energy effective action of stringtheory and because it has proven useful in performing two-dimensional semi-classicalcalculations of black hole evolution. The simplest example of such a theory, as discussedin Appendix B, is given by

�dilaton = (-g)1/2 e!2M (R + 4(LM)2 ! F:<F:<),

where the dilaton, M, is a scalar field.

75

where g is the determinant of the 4x4 matrix built out of the components of the metric

tensor, g:<, and R is the scalar formed by contacting the Ricci tensor, R:<. If there are

matter fields present, we add their Lagrangians to find our equations of motion; that is,

we extremize the action derived from the total Lagrangian � = �E + �matter. For example,

adding the Lagrangian density for Maxwell’s equations, �M = F:<F:<, yields general

relativistic electromagnetism.2 We find that the equations of motion couple the stress

energy tensor, T :<, of the matter fields to the metric according to the Einstein field

equations:

R:< ! ½ g:<R = 8BT :<, (3.1)

where Newton’s constant, G, and the speed of light, c, have been set equal to one.

The first discovered solution to the vacuum field equations was the Schwarzschild

solution, which describes a static, spherically symmetric, matter-free spacetime. The

metric for this spacetime is given by

ds2 = !(1 ! 2M/r) dt2 + (1 ! 2M/r)!1 dr2 + r2 dS2, (3.2)

where r is the radial coordinate and dS2 is the metric of a two-sphere, d2 2 + sin22 dN 2.

In this vacuum solution, M is an arbitrary parameter; however, as indicated by the

notation, it is naturally associated with mass. This is because the Schwarzschild solution

also describes spacetime outside a spherically symmetric distribution of matter with total

76

mass M. If this matter is collapsing down – e.g., if it is undergoing gravitational collapse

– then the Schwarzschild solution will be isometric to more and more of the spacetime.

As can be seen from Equation (3.2), the Schwarzschild solution is singular at r = 0

and at r = 2M. These two singularities are importantly different, however. The central

singularity is a true singularity of the solution as can be seen from the fact that curvature

invariants (i.e., scalar invariants constructed from the components of the Riemann tensor)

blow up as we approach r = 0. However, these invariants are well behaved at the so-

called Schwarzschild radius, r = 2M. Indeed, by using an appropriate coordinate

transformation we can extend our description of the spacetime beyond this region. We

shall have more to say about this maneuver below in Section 3.3, but for now it will

suffice to exhibit a choice of coordinates that removes the singularity. We can define

Kruskal coordinates (X, T) according to the following equations:

(r/2M !1) e r / 2M = X 2 ! T 2 ,

t /2M = ln[(T+X)/(X!T)] . (3.3)

In these coordinates the metric (3.2) can be reexpressed as

ds2 = (1/r) 32M 3 e r / 2M (!dT 2 + dX 2) + r2 dS2.

In this form it is clear that the metric is smooth at r = 2M. The singularity at r = 0, on the

other hand, cannot be removed by any coordinate transformation, and is thus an intrinsic

singularity of the spacetime.

Although it is not a singularity, the Schwarzschild radius is nevertheless a

significant feature of this spacetime. Any causal – i.e., luminal or sub-luminal –

maximally extended curve that passes within r = 2M will terminate at the central

77

Figure 3.1: Conformal spacetime diagram of black hole.

singularity in a finite distance as measured by its affine parameter. This implies that the

region within r = 2M is not in the causal past of future null infinity, that is, light signals

from this region cannot escape to infinity.

A Penrose diagram of Schwarzschild spacetime is given in Figure 3.1. (For an

explanation of Penrose diagrams and structure of infinity depicted therein, including

future null infinity I+, see Appendix A.) The shaded region of the diagram is the black

hole which is defined as the region of spacetime that is not in the casual past of future

null infinity, I+. Thus, for Schwarzschild spacetime, the region within r = 2M is a black

hole. Any causal trajectory through this region terminates in the singularity indicated by

the jagged line. The border of a black hole is referred to as the event horizon, which will

3 See, for example, the historical overview in Thorne et al. (1986) pp. 1-5.

78

be a null path through the spacetime. It is the path of the last light front that succeeds in

escaping the black hole singularity.

One can glimpse several important features of the event horizon from this

defining role that it plays as the border between outgoing light paths that escape to

infinity and those that terminate in the singularity. The first is that the location of the

event horizon is a global concept, depending on the structure of the spacetime as a whole.

We therefore do not expect a freely falling observer to notice anything unusual at the

event horizon of a large black hole. The second point is that an external observer will

never see anything that occurs beyond the horizon; thus such an observer will never see

any infalling object actually cross the event horizon. If the external observer uses the

usual Einsteinian half-light-trip simultaneity convention (which will yield Schwarzschild

coordinates if he remains at fixed r) he will find that all processes slow down and

eventually freeze as an object approaches the event horizon. For this reason, stars that

underwent total gravitational collapse were generally called “frozen stars,” or “collapsed

stars,” until the mid-1960s.3

Although this frozen-star picture is now often scorned as an extremely limited

viewpoint that relies too heavily on a particular choice of coordinates (namely

Schwarzschild coordinates), it can offer an accurate, and useful, description of physics

outside a black hole. This viewpoint has been dubbed the “membrane paradigm” of black

holes (see Thorne et al., 1986). The physics of this viewpoint is given in a universal time

t, which is taken to be the Schwarzschild time. This requires a 3+1 split of the four-

79

dimensional spacetime metric g:< into a three-dimensional purely spatial metric, gab, and

the universal time coordinate, t. The observers for whom this description would be

appropriate are fiducial observers who remain at rest a fixed radial distance from the

black hole.

From this perspective all physical processes slow down as one approaches the

event horizon, and at the horizon physics freezes completely in the universal time. Our

physical description will therefore only be valid outside the black hole and will require us

to impose appropriate boundary conditions at, or outside, the event horizon. To keep the

relevant quantities finite, one defines a “stretched horizon” outside of the true event

horizon. The proper boundary conditions will be imposed at the stretched horizon –

which can thereby be attributed a mass, charge, surface current and resistivity, and other

properties. From the external observer’s perspective, a black hole will behave like a

physical membrane situated at the stretched horizon.

The accuracy of the membrane description will depend on the distance between

the stretched horizon and the true horizon. For astronomical purposes, nearly all of the

near horizon details are irrelevant, and thus a description of a membrane at a macroscopic

distance from the event horizon offers a sufficiently accurate description of processes in

the vicinity of the black hole. We can develop a more precise description, to any order of

(classical) accuracy, by choosing the stretched horizon to be closer to the true horizon.

As we shall see in Chapter 5, some advocates of black hole complementarity argue that a

legitimate quantum description of a black hole can be developed by considering a

stretched horizon whose area is only one Planck unit (10!70 m2) larger than the area of the

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event horizon. Such a level of accuracy is, of course, far beyond the interests of most

astrophysicists.

We shall not be concerned with the particular details of the description of the

membrane at the stretched horizon. The important point is that the membrane is not

merely a spacetime surface at which we impose certain boundary conditions, but it is an

“object” whose evolution is described by familiar physical laws, such as Maxwell’s

equations. This allows us, for example, to consider a black hole as a resistor in an electric

circuit or a rotor in an electric motor (Thorne et al. 1986, pp. 52-57).

The membrane description of black holes misrepresents, or perhaps is silent

about, the region of spacetime inside the stretched horizon. Thus this description would

be inappropriate for an observer freely falling into a black hole. Such an observer would

notice nothing out of the ordinary at the stretched horizon, and would not notice a

membrane, or any other signature of a black hole horizon. The apparent incompatibility

of the freely falling observer and the fiducial observer is, at this point at least, an

unproblematic relativistic effect. The slowing of physics at the horizon observed by the

fiducial observer, but not the infalling observer, is merely an example of time dilation

effects familiar from special and general relativity. As we shall see in Chapter 5,

however, black hole complementarians argue that when we move to a quantum

mechanical description of the black hole the relationship between these two perspectives

becomes more problematic

81

3.2: Black Hole Mechanics

In the early 1970s a striking similarity between the standard laws of

thermodynamics and the evolution of black holes began to emerge. The earliest parallel

to be noticed was that between entropy and the surface area of a black hole. Bekenstein, a

student of Wheeler’s at the time, argued that the second law of thermodynamics requires

one to assign a finite entropy to a black hole. We shall look at this issue more carefully in

Section 3.6, but the gist of the worry is that one could collapse any amount of highly

entropic matter into a black hole – which is apparently an extremely simple object

described by a mere handful of parameters – leaving no trace of the original disorder.

Bekenstein’s conviction that the area of a black hole really is a measure of entropy was

strengthened by Hawking’s 1972 proof of the area increase theorem, which states that

(classically) the surface area of a black hole, like the entropy of a closed system, can

never decrease.

The similarity between the two cases was considerably strengthened when

Bardeen, Carter, and Hawking (1973) proved three other laws of black hole dynamics that

exactly parallel the first, third, and “zeroth” law of thermodynamics. These laws were

originally derived for stationary black holes satisfying the dominant energy condition, but

further research has shown that they hold more generally.

The laws of black hole mechanics are formulated by considering stationary

spacetimes, which are spacetimes that contain a one parameter group of timelike

isometries, that is, maps (more specifically, diffeomorphisms) that leave the metric g:<

invariant. The infinitesimal coordinate transformation of such a group of isometries is

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called a “Killing vector field.” This vector field will allow us to define the surface

gravity of a black hole, which is crucial for stating the laws of black hole mechanics.

A Killing horizon, which we will label hK, is a null surface whose generators

coincide with the orbits of a Killing field; the event horizon of a stationary black hole is

an example of such a horizon, as is the horizon of Rindler spacetime, which we shall

discuss below. By definition, there is a Killing field, P:, normal to hK, but this need not

be the same field that generates the timelike isometries of the spacetime, call it .:.

However, the two are related according to

P: = .: + ShR:.

Here R: is an axial Killing field and Sh is referred to as the angular velocity of the

horizon. A rotating black hole (i.e., a Kerr black hole) has a nonzero angular velocity.

The Killing field P: will be normal (as well as tangent) to hK. Also normal to hK is

L:(P<P<), since P<P< has the constant value of zero along this null surface. This implies

that the two will be proportional to one another, and we can define the function 6 on the

horizon according to

L:(P<P<) = !26P:.

The function 6 is called the surface gravity of the black hole because it is the limit of the

force that an observer at infinity would have to exert to hold a test particle of unit mass

stationary as it approaches the event (Killing) horizon (by a massless, arbitrarily strong

rope). If we consider the region outside the horizon we can define a, which will be the

magnitude of acceleration felt by an observer following the trajectories of P:, as

a = (a:a:)1/2, where a: = (P<L<P

:)/(!P8P8). Using the redshift factor W, defined as

4 For a proof of these laws of black hole mechanics see Bardeen et al. (1973), or Wald(1984) pp. 330-337.

83

W = (P:P:)1/2, the surface gravity 6 can be written as

6 = limr6h(Wa) (3.4)

where the limit is taken as one approaches the horizon h.

The acceleration experienced by a fiducial observer outside a Schwarzschild black

hole is given by a and blows up as one approaches the horizon. However, the redshift

factor W implies that an observer at infinity would only need to exert a finite force, given

by 6, to suspend a unit mass infinitesimally above the horizon. Once the mass reaches

the horizon, of course, the local force needed to keep the mass stationary goes to infinity,

and the rope has to break.

The zeroth law of black hole mechanics states that for a stationary black hole 6 is

constant over the entire event horizon.4

The first law compares two Schwarzschild black holes whose mass differs by the

infinitesimal amount *M. The difference in the area of the event horizon that

accompanies this difference is *A, given by

*A = (8B/6) *M.

More generally, we can also consider the angular momentum, J, of a black hole and the

angular velocity of the horizon, Sh, defined above. We then find

*M = (1/8B) 6 *A + Sh *J.

This, of course, is remarkably similar to the first law of thermodynamics:

*E = T *S ! P *V,

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where this equation relates the energy (E) of a system to its temperature (T), entropy (S),

pressure (P), and volume (V).

The second law derived by Bardeen et al. states that the total surface area of a

black hole can never decrease. This law is based on the dominant energy condition which

is generally violated by quantum fields in curved spacetime. Once we allow such fields

this law can be broken, and we expect Hawking radiation to remove mass from the black

hole and thus for A to decrease over time. However, Bekenstein has argued that there is a

generalized second law of thermodynamics, which states that if one adds together the

total entropy of a system and surface area of any black holes present, this number can

never increase. We will discuss this suggestion in more detail below.

The third law of black hole dynamics states that one can never reach 6 = 0 in any

finite process. This is analogous to the “Nernst formulation” of the third law of

thermodynamics, which states that it is impossible to reach absolute zero by any finite

process. The analog of the “Planck formulation” of the third law, which says that S goes

to zero as the temperature T goes to zero, is not satisfied by black hole mechanics (since,

for example, 6 = 0 for all extremal black holes, but A /=0). However, Wald (1994, p. 148),

for example, argues that one should not think that this formulation of the third law is

fundamental, as it can be violated in some circumstances.

The symmetry between these laws and the laws of thermodynamics is all the more

striking in view of the relativistic equivalence of mass and energy. Bekenstein took this

as strong confirmation of his earlier conjecture that the area of a black hole really is a

measure of its entropy. However, equating the two sets of laws required one to assign a

5 We can motivate this transformation, as well as the next one, by looking at the affineparameterization of the geodesics of the spacetime. Such arguments can be found inWald (1984), pp. 149-153, whose treatment I am closely following here.

85

temperature to a black hole, which implies that a black hole must radiate. Since everyone

involved agreed that this was impossible, the parallel between the two was not taken very

seriously. This changed in 1974 with Hawking's monumental discovery that even a

stationary black hole will give off radiation. In preparation for a discussion of this

derivation, we first consider Rindler spacetime and the Unruh effect.

3.3: Rindler Spacetime and Coordinate Limitations

A helpful analogy for understanding many features of a black hole spacetime is

offered by Rindler spacetime, given by the metric

ds2 = !>2 d02 + d>2, (3.5)

where for simplicity we limit ourselves to a two-dimensional spacetime. The components

of the inverse metric g:< blow up at >=0, implying that the spacetime is singular here.

However, through a judicious coordinate transformation or two, we can rewrite the metric

in a form that removes the singularity at >=0, and allows us to extend our description of

the spacetime into regions where (3.5) breaks down.

To see this, let us first define the null coordinates5

u = 0 ! ln >, (3.6)

v = 0 + ln >.

In these coordinates the metric of Rindler spacetime is given by

ds2 = !ev!u dudv. (3.7)

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These coordinates are still limited to >>0, of Rindler spacetime, and so we have not yet

succeeded in finding coordinates that allow us to extend the spacetime beyond the

(coordinate) singularity at >=0 (corresponding to u = 4 and v = !4). We therefore define

a further set of coordinates:

U = !e!u, (3.8)

V = e v,

in terms of which the metric takes the form

ds2 = !dUdV. (3.9)

This equation is the Minkowski metric written in lightcone coordinates, which are

defined as

U = t ! x, (3.10)

V = t + x.

This can easily be seen by substituting the coordinates (t, x) into (3.7), yielding the

familiar form of the Minkowski metric:

ds2 = !dt2 + dx2. (3.11)

We therefore see that the metric of Rindler spacetime is identical to that of

Minkowski spacetime – at least where it is well defined. However, the definition of

Rindler spacetime given in (3.5), and in (3.7), is limited to >>0; thus Rindler spacetime is

only part of Minkowski spacetime. When we transform the metric into lightcone

coordinates, (3.9), we should note that the two metrics are only equivalent if we restrict

(3.9) to U < 0 < V. Thus Rindler spacetime is equivalent to a wedge of Minkowski

spacetime given by x > |t|; we have labeled this wedge R in Figure 3.2.

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Figure 3.2: Rindler Spacetime

Paths of constant > through Rindler spacetime will represent paths of constant

acceleration in Minkowski spacetime, where the acceleration a is given by 1/>. This

means that an observer who is following such a trajectory will be experiencing an

acceleration of a. If such an observer were to establish planes of simultaneity by sending

out light signals and recording the time that it took them to return from some point (i.e.,

using the Einstein simultaneity convention), these planes of simultaneity would be planes

of constant 0. Thus Rindler coordinates can be viewed as a set of coordinates that a

constantly accelerating observer in Minkowski spacetime would naturally use. Notice

that such an observer could not be affected by any events in region L or F. Region L is a

Rindler spacetime in its own right, where here the lines of constant > correspond to left-

moving, constantly accelerating observers. Such observers are cannot interact with

regions R or F.

88

The null surfaces labeled hR and hL are horizons for right-moving and left-moving

Rindler observers (as we shall call them). No signal (propagating at or below the speed

of light) that originates to the left of hR, for example, can ever reach the constantly

accelerating observer in R. It is not hard to see that (3.5) is symmetric under translations

in 0. This timelike isometry is generated by a Killing vector field, P:, that in the full

Minkowski spacetime is simply the generator of boosts. Given that Minkowski

spacetime is isometric under boosts, and that the Killing vector field, P:, vanishes on hR

and hL, we can see that these two surfaces are also Killing horizons. As such they can

fruitfully be compared to the event horizon of a stationary black hole.

Let us pause for a moment to look more carefully at the process, described above,

of “removing the coordinate singularity” at >=0. While the spacetime defined in (3.5) is

only valid for >>0, once we make the transformation to (3.9) we have an easy prescription

for extending the spacetime into the full Minkowski spacetime: We simply remove the

requirement that U and V be restricted by U<0<V. That is, even though Rindler

spacetime ends at >=0, we can consider a new spacetime that does not end here, but

continues smoothly through this surface. Rindler spacetime is identical to (is a

redescription of) part of Minkowski spacetime, and the full Minkowski spacetime is

identical to the maximally extended Rindler spacetime.

Let us now compare this situation to the case of Schwarzschild spacetime. In

Section 3.1 we dismissed the singularity at r = 2M as a “mere” coordinate singularity.

However, if we take equation (3.2) to define a spacetime (as opposed to picking out a

spacetime by some part of it, for example) then it seems we should say that

6 This decomposition assumes that the space, call it ,, of “positive frequency” solutionsfi(j) satisfies the following three conditions (see Wald 1994, p. 27): (i) the inner product

is positive definite on ,, i.e., 0 # (f(i), f(j)) < 4; (ii) the total (complexified) solution

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Schwarzschild spacetime actually ends at r = 2M. Of course, we can once again shift into

coordinates that allow us to extend the metric into a new spacetime, of which

Schwarzschild spacetime will be a part – just as Rindler spacetime is a part of Minkowski

spacetime. In our new coordinates we can do this by simply removing the “artificial”

limitations imposed by our previous “poorly chosen” coordinates. We will continue to

follow standard practice in referring to the extended spacetime as the Schwarzschild

solution, that is, this will designate the full geometrical object, and not merely the part of

spacetime that lies outside r = 2M.

3.4: The Unruh Effect

In Section 2.1 we were able to construct the Hilbert space for a scalar field by

identifying the positive frequency solutions, ek, to the equations of motion and then using

them to decompose a general field operator N(x) according to Equation (2.3). In this

section we shall follow Birrell and Davies (1982) pp. 113-115, and write our

decomposition as the discrete sum:

N(x) = Gk (âk ek + âk† ek*). (3.12)

As we saw, this decomposition allows us to define our vacuum state |0,, and construct our

Fock space from it by successively applying creation operators âk†.

We can also consider other decompositions of the field based on a different

complete set of “positive frequency” modes,6 say fj(x),

space S÷ is spanned by , and its complex conjugate space ,) ; (iii) (f(i), f(j)*) = 0, for allf(i)0, and all f(j)*0,) .

7 The two theories can be shown to be unitarily equivalent if two conditions hold, namelythat the real valued inner products associated with these two choices of subspaces ofsolutions define equivalent norms, and that the trace $jk

* $kj is finite. If the two theoriesare unitarily equivalent then one can show that (jk = "jk

* and *jk = !$ij. See Wald (1994)p. 71 for details of these two conditions for unitary equivalence, and for proofs mentionedin the following paragraph. Recall that a QFT is defined by a Hilbert space and a set ofoperators on that space (,, Ôi). Two “theories” are unitarily equivalent if and only ifthere is a unitary operator U: ,6,N such that for all Ôi there is an ÔiN in ,N given by UÔi U

!1 = ÔiN.

90

N(x) = Gj [ b$j† fj + b$j f

j* ]. (3.13)

This decomposition allows us to define a new vacuum, |0b,, and a new Fock space based

on that vacuum and the creation operators b$j†.

The relationship between the quantum field theory defined by this construction,

and our previous QFT will depend on the relationship of these two decompositions of the

field. We can express one set of modes in terms of the other according to the following

relations, known as the Bogoliubov transformations:

fj(x) = Gk ("jk ek + $jk ek*), (3.14)

ek = Gj ((jk fj + *jk fj*).

The coefficients "jk and $jk are referred to as the Bogoliubov coefficients.7

The Bogoliubov coefficients can be used to characterize the transition amplitudes,

that is the S-matrix, between states in the representations of the Hilbert space based on the

two different vacuums. The number operator for the new Fock space, Nj = b$ j†b$ j, will

generally have a nonvanishing expectation value for the vacuum, |0in,, of the old theory.

This expectation value is given by

91

+0in| Nj |0in, = Gk |$jk|2,

and thus will be nonzero if $jk =/ 0. Thus in this new theory the old vacuum will contain

particles. We can see from (3.14) that this particle content is tied to fact that the positive

frequency solutions of the new theory are a combination of the negative as well as the

positive solutions of the old theory.

Although we can in principle pick any decomposition of the space of solutions

that satisfies the conditions mentioned in Footnote 6, in practice choosing a subspace of

positive frequency solutions depends on finding an appropriate time coordinate in our

spacetime. If, for example, the spacetime contains a timelike Killing vector field, we can

use the time coordinate associated with that field to pick out our positive frequency

solutions. In Minkowski spacetime we have two distinct choices of timelike Killing

fields, and thus two natural choices of time coordinates: the standard time coordinate of

an inertial observer, and that of an accelerated Rindler observer. More generally, we will

be able to pick out a preferred time coordinate in any stationary spacetime, for by

definition such a spacetime possesses a timelike Killing field. We can also consider

spacetimes whose metric asymptotically approaches the metric of a stationary spacetime

in the distant past and/or future.

To derive the Unruh effect we find the Bogoliubov coefficients that relate the

QFTs associated with the Minkowski observer and the Rindler observer and then show

that the Minkowski vacuum will be thermally populated with particles characterized by

the Rindler creation operators.

8 There is some debate over the legitimacy of this argument. Arageorgis, Earman, andRuetsche (unpublished) point out that this surface does not include the point at the originand thus is not a Cauchy surface. The relevance of this observation seems questionable,however, given that one can instead define the relevant modes on the null surfaces hR andhL, reaching the same result (see Wald 1994, p. 110-114). I shall therefore continue tofollow the relatively straightforward account of Birrell and Davies. We shall return to thequestion of the robustness of this derivation, and that of Hawking radiation, below.

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We now consider the Unruh effect for a massless scalar field. Following Birrell

and Davies (1982) p. 113, we begin by rewriting the Klein-Gordon equation in Rindler

coordinates as

(M/M02 ! M/M>2) N = 0.

We can then define solutions to this equation that will be purely positive frequency in the

right hand Rindler wedge (R) and vanish in the left hand wedge (L):

f R(p) = (4BT)!1/2 eip> ! iE0 in R

= 0 in L,

where E = |p|. In L, the Killing vector field is oriented in the opposite direction (i.e.,

“pastward”). Thus the positive frequency solutions for this region are given by

f L(p) = (4BT)!1/2 eip> + iE0, in L

= 0 in R.

Each of these sets is complete for its respective Rindler spacetime. We can

consider a surface of constant 0 across both L and R to be a Cauchy surface, which

allows us to analytically extend the modes from L and R into the future and past wedges

of Minkowski spacetime.8 Together, the f L(p) and f R(p) offer a complete set of modes

93

for the Klein-Gordon equation for all of Minkowski spacetime. Thus we can decompose

our field operators according to

N = Gp [b$pR f R(p) + b$p

R† f R*(p) + b$pL f L(p) + b$p

L† f L*(p)]. (3.15)

Now we need to determine the Bogoliubov transformation between this set of

modes and that of an inertial Minkowski observer. To do so we compose from the

Rindler modes a set of functions that are purely positive frequency with respect to inertial

Minkowski time.

F1 = [eBE/a f R(p) + e!BE/a f L*(!p)]/[2 sinh(BE/a)]1/2

F2 = [e!BE/a f R*(!p) + eBE/a f L(p)]/[2 sinh(BE/a)]1/2

where a is the acceleration of our Rindler observer. Let us now decompose N with

respect to these functions

N = Gp [d$1 F1 + d$2 F2 + d$1† F1

* + d$2† F2

*]. (3.16)

Because these modes share the inertial Minkowski observer’s characterization of positive

frequencies, the annihilation operators d$1 = d$1(p) and d$2 = d$2(p) share a common vacuum

state with the Minkowski representation (3.4); i.e.,

d$1|0M, = d$2 |0M, = 0.

Thus we can use these new annihilation operators to characterize the Minkowski vacuum.

We now consider the inner product (N, f R(p)), first using the Rindler expansion

(3.15) which simply yields b$pR. Next we compute this inner product using the modified

Minkowski decomposition given in (3.16), where we pick up a term containing d$1 and

one containing d$2†. This, and a similar comparison of the two decompositions of

(N, f L(p)), give us

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b$pR = [eBE/2a d$1 + e!BE/2a d$2

†] / [2 sinh(BE/a)]1/2

b$pL = [eBE/2a d$2 + e!BE/2a d$1

†] / [2 sinh(BE/a)]1/2.

We now have the resources to characterize the particle content that an accelerating

observer will observe in the Minkowski vacuum state. This Rindler observer’s number

operator (for particles of momentum p) is given by

N$ pR = b$p

R† b$pR .

We therefore find that the expectation value for the number of particles that such a

constantly accelerating observer will detect is the following:

+0M | N$ pR | 0M, = [2 sinh(BE/a)]!1 +0M |(eBE/2a d$1

† + e!BE/2a d$2)(eBE/2a d$1 + e!BE/2a d$2

†)| 0M,

= [2 sinh(BE/a)]!1 +0M | e!BE/2a d$2 e!BE/2a d$2

† | 0M,

= [2 sinh(BE/a)]!1 e!BE/a

= 1/(e2BE/a !1).

This is the spectrum for thermal radiation at temperature

T = a/2BkB. (3.17)

This is then the ground for claim that an accelerating observer will experience a thermal

bath of particles at a temperature proportional to her acceleration.

3.5: Hawking Radiation

The derivation of Hawking radiation follows the derivation of the Unruh effect

very closely, both formally and conceptually. Here we consider a black hole formed by

the collapse of a spherically symmetric shell of matter. (This is depicted below in Figure

3.3.) The metric outside the shell will be the Schwarzschild metric, while the metric

9 This derivation follows Fré et al. (1999), pp. 47-51, and Hawking (1975).

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inside the shell will the Minkowski metric (although the analysis below will also apply

for an arbitrary interior metric).

Because our spacetime is spherically symmetric we can decompose an arbitrary

solution to the Klein-Gordon equation as9

N(x) = r !1 f(t,r) Ylm(2,N)

where Ylm(2,N) are spherical harmonics. If we use the tortoise coordinate r* , defined as

r* = r + 2M ln|1 ! r/2M |,

we find that our massless Klein-Gordon equation in the Schwarzschild metric yields

!M2f /dt2 + M2f /d r*2 + V(r*)f = 0 (3.18)

where V(r*), which can be seen as a potential term, is given by

V(r*) = !(1!2M/r) [l(l+1)/r2 + 2M/r3]. (3.19)

To solve (3.18) we need to determine the appropriate boundary conditions. We see that

as we approach the horizon, r = 2M, the potential (3.19) vanishes, and likewise V(r*) 6 0

as r 6 4. Thus as we approach either the horizon or spatial infinity, (3.18) approaches a

free wave equation. Therefore, positive frequency solutions near the horizon will

approach plane wave solutions in tortoise coordinates (t, r*):

fT(t, r*) = b e!iTt eiTr* + c e!iTt e!iTr*.

We now change to the null coordinates u = t!r*, v = t+r*, which can be compared to the

transformation from Rindler coordinates given in (3.6), and define:

fT(out) = b e!iTu ,

fT(in) = c e!iTv.

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This allows us to write our positive frequency solutions as

fT = fT(out) + fT

(in).

Still paralleling the Rindler case (cf. (3.8) above) we recast fT(out) in terms of the

coordinates (U, V) defined as U = !e!u/4M; V = ev/4M :

fT(out) = b e!iT 4M ln(!U) = b e!iT ln(!U) /6,

where this last simply follows from the fact that the surface gravity 6 is given by 1/4M.

The outgoing modes fT(out) will oscillate very rapidly near the horizon. Indeed,

because U 6 4 as r 6 2M, the frequency of these modes will diverge as we approach the

horizon. (We shall see in Chapter 5 that this has led some to question the validity of the

semi-classical approximation in this region.) That is, if we are considering extremely

energetic modes of the field (which, as we shall see momentarily, are the modes that are

detected as Hawking radiation) then we might question whether we can safely neglect the

effect of these modes on the metric and on the infalling matter. At the moment, however,

we only need to recognize that the high frequency of these modes means that geometric

optics should be extremely accurate in this region, and we can consider these modes to be

rays coming in from I! and out to I+.

We consider an outgoing ray at I+ and trace it back to the infinite past. At the

point that the ray encounters the collapsing matter the metric changes, and a portion of the

ray is reflected back to I! while another portion passes through the matter before reaching

I!. This second portion is the part we will want to pay attention to in our derivation.

Consider the ray labeled ( in Figure 3.3. Because the spacetime is spherically symmetric

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Figure 3.3: Outgoing mode in a black hole spacetime.

about r = 0 the ray from I! to I+ is depicted as reflected the origin. The latest ray that

reaches I+ travels along the event horizon hS, and we label the continuation of this ray

back to I! by (h.

If we choose our coordinates so that v = 0 at (h, then we see that in the

neighborhood of v = 0 on I!, we will have fT(v) = 0 for v>0, for no modes propagated

back from I+ through the collapsing matter can reach I! after this point. For v<0 we have

fT(v) = fT(out) = b e!iT ln(!v)/6. (3.20)

This, recall, is our solution that is positive frequency on I+.

We now wish to discover the Bogoliubov coefficients that relate this solution to

solutions that are purely positive frequency on I!, where such solutions are given by

e#TN(v) = (TN)!1/2 e!iTNv.

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The relevant Bogoliubov transformation will then be

fT(v) = I04 dT ("TTN e#TN(v) + $TTN e#TN(v)*), (3.21)

and we have

$TTN = ( fT(v), e#TN(v)) = Idv c e!iT ln(!v)/6 (TN)!1/2 e!iTNv. (3.22)

Our next step in evaluating this is to consider the Fourier transform of fT(v) with respect

to the frequency TN, that is, we define

f#(TN) = (1/2B)1/2 Idv eiTNv fT(v),

where this integral need only run from !4 to zero, because fT(v) = 0 for v>0. Inserting

(3.20) into this equation gives us,

f#(TN) =

From this we can establish that the relationship between the negative frequency functions,

f#(!TN), and the positive frequency functions, f#(TN), must obey the relation

f#(!TN) = !eTB/6 f#(TN).

Thus we express our original functions that are positive frequency on I+ as,

Comparing this with equation (3.21) we see that e#TN(v) % f#(TN)e!iTNv and that our

Bogoliubov coefficients must satisfy the relation

$TTN = !e!iTB/6 "TTN.

This, together with (3.21), allows us to solve Equation (3.22) in terms of '-functions (see

Birrell and Davies, p. 108), from which we find

10 It is worth noting that although the spectrums of both Hawking radiation and the Unruheffect are thermal, there are modes populated in the case of the Unruh effect that areunpopulated in Hawking radiation. See Wald (1994) p. 155 for details.

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|$TTN|2 = [(2B6TN)(exp(T2B/6)!1)]!1.

Although the total number of particles produced diverges – because the black hole

emits particles for all future time – we can calculate the (finite) number of particles per

unit time either by utilizing wave packets or by confining the system to a box. For a

frequency range from T to T+dT this yields a number of particles per unit time given by

(Birrell and Davies, p. 260)

(dT/2B)(e2BT/6 ! 1),

which again is the Planck spectrum with a temperature of

T = 6/2BkB. (3.23)

Comparing this with (3.17) we see that the surface gravity 6 plays the same role in

Hawking radiation that the Rindler observer’s acceleration a played in the case of the

Unruh effect.10

A couple of points should be made about the radiation predicted in this derivation.

The first is that the Hawking effect is a purely kinematical effect; our field is a free field

that neither interacts with itself nor with the matter forming the black hole. The radiation

is purely a result of the geometry of the spacetime. The second point is that, despite the

idealizations made in the above derivation, the prediction of Hawking radiation seems to

be quite robust. The effect can be derived for more general types of black holes, for

interacting fields, and using other techniques, e.g., the path integral formulation of QFT.

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The robustness of this derivation is also supported by Unruh’s investigations (e.g.,

in Callender and Huggett 2001) of “dumb holes,” the sonic analogues of black holes. We

can consider sound waves in a medium that is allowed to flow. If the rate of flow exceeds

the speed of sound, we will have a sonic horizon out of which no sound can escape. If we

quantize this theory, we find that the dumb hole gives off radiation analogous to Hawking

radiation. The significant point is that the existence of this sonic radiation appears to be

independent of the fine-grained details of the medium. That is, the prediction seems not

to depend on the size or detailed behavior of the molecules making up our fluid. This

strongly suggests that Hawking radiation is likewise largely independent of the short-

wavelength (high energy) behavior of the field. That is, even if our field is composed of

“molecules” whose size establishes the limits of our field theoretic description, we should

still expect Hawking radiation upon quantizing the field.

Our derivation of (3.23) gives us the temperature of Hawking radiation for an

observer at spatial infinity. We can also consider the temperature that would be

registered by detectors closer to the black hole, carried either by fiducial observers or

freely falling observers. The radiation at infinity has undergone a redshift in climbing out

of the gravitational well. This redshift factor W is given by

W=(P:P:)1/2, (3.24)

where P: is the Killing vector field that defines the timelike isometry of our static

spacetime (cf. (3.4) above). In Schwarzschild spacetime we have W(r) = (1 ! 2M/r)1/2.

The temperature that will be felt by an inertial observer at radius r will then be given by

T(r) = 6/2BkBW(r).

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We now recall that according to the definition (3.4) the surface gravity 6 of a

Schwarzschild black hole is given by 6=limr62M (aW), where a is the acceleration of a

fiducial observer. Thus, as we approach the horizon, we see that the temperature of the

Hawking radiation approaches

T(r62M) = a/2BkB,

which is just the Unruh temperature given in (3.17) for temperature of particles

experienced by an observer with acceleration a. Thus it appears that near the horizon, the

detection of Hawking radiation by fiducial observers can be seen merely as a

manifestation of the Unruh effect for observers undergoing acceleration to remain a fixed

distance from the event horizon. This is one way of recognizing that a freely falling

(unaccelerated) observer will see the field in its vacuum state, just as the inertial observer

in Minkowski space sees a vacuum where the Rindler observer sees a temperature

proportional to a. Given that the fiducial observer is equivalent to the accelerating

Rindler observer, the infalling observer will be equivalent to the Minkowski observer;

that is, she will detect none of the radiation effects.

3.6: The Generalized Second Law

As we saw above, in the early 1970s a remarkable similarity between the laws of

the evolution of black holes and the laws of thermodynamics was noticed. The parallel

laws are laid out in a table in Figure 3.4. Although there was considerable skepticism

over the significance of these parallels at first, the discovery that black holes could

apparently be assigned a physically meaningful temperature led many physicists to

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Black Hole Mechanics6 is uniform over event horizon.

*M = (1/8B) 6 *A + Sh *J.A never decreases with time.

6=0 cannot be reached.

0th Law1st Law2nd Law3rd Law

ThermodynamicsT is uniform for body in equilibrium.

*E = T *S ! P *V.S never decreases with time.

T=0 cannot be reached.

Figure 3.4: Laws of black hole mechanics and thermodynamics.

believe that the connection between these two sets of laws might be quite substantial.

The identification of mass with energy was, of course, already unproblematic. Bekenstein

had suggested on general thermodynamical grounds that the area of a black hole should

be identified with entropy. Now the most significant missing link, and the most

questionable on the face of it, the link between surface gravity and temperature, had been

established.

The original worry that confronted Wheeler and Bekenstein was the question of

whether one could violate the ordinary second law by dumping high-entropy (e.g.,

thermal) material into a black hole. A black hole is a very simple object – specified only

by its mass, angular momentum, and charge. Thus we might be concerned in some

general way about the apparent disappearance of disorder from the universe – as this

seems to threaten the second law. We shall investigate a more rigorous objection below,

based on the apparent possibility of using a black hole to convert all of the thermal energy

in a reservoir into work.

These concerns over the ordinary second law led Bekenstein to suggest that a

black hole should be assigned an entropy, but this led to the question of how much

entropy a black hole should have. We might be tempted simply to assign it the total

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entropy of all the matter that fell into it. However, such a designation would be

somewhat questionable as there would be no way to have the swallowed matter interact

with other thermodynamical systems: there is no difference (on the outside) between a

black hole that has swallowed thermal radiation and one that has been fed exclusively on

pure laser light. Once Hawking discovered how to assign a precise temperature to a

given black hole, however, it was straightforward to infer the entropy that should be

associated with it. From (3.23) we have TBH = 6/2BkB, which gives us

SBH = kB A/4, (3.25)

if we disregard the additive constant that can generally be included when defining

entropy.

The discovery of Hawking radiation seemed to place the second law of black hole

mechanics in jeopardy, because the back reaction of the radiation on the metric would

presumably result in the loss of mass by the hole, and thus the shrinking of its surface

area. However, the evaporation of a black hole results in thermal radiation, which raises

the thermodynamic entropy of the system. Thus, although individually the ordinary

second law and the second law of black hole mechanics can be violated, there should be a

generalized second law stating that the sum of the total thermodynamic entropy and the

total black hole entropy can never decrease in any closed system. That is, we define a

generalized entropy by

SG = STD + SBH, (3.26)

and propose that this generalized entropy can never decrease with time.

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It is instructive to see that this claim is not trivial – and that a fairly strong case

can be made for its truth. Consider a box of thermal radiation that is slowly lowered

towards the event horizon of a black hole by an observer arbitrarily distant from the black

hole (i.e., “at infinity”). This observer can extract work from the energy in the box by

lowering it in the gravitational field of the black hole. Once the box is near the event

horizon, the bottom of the box is opened and the radiation is released into the black hole.

The considerably less massive box (as an idealization we can consider it and the rope to

have vanishing mass) can then be drawn back out to infinity and refilled with thermal

radiation for another cycle.

Now consider the amount of work that such a process can produce. For the

observer at infinity the energy of the radiation, E, will be redshifted according to

, = W(r) E, (3.26)

where W(r) is the redshift factor defined in (3.24) and for a Schwarzschild black hole is

given by,

W(r) = (1 ! 2M/r)1/2. (3.27)

Thus , decreases as the box approaches the event horizon, and can apparently be made

arbitrarily small by allowing the box to approach nearer and nearer to the event horizon.

The work that can be extracted by lowering the box is given by the difference of E and ,,

which implies that all of the energy of the thermal radiation (excepting the apparently

arbitrarily small ,), can be extracted from the radiation in the box. By refilling the box

from a thermal resevoir, it appears as though all the thermal energy in a heat source can

11 This version states that it is impossible to construct a device that, operating in a cycle,can extract heat from a reservoir and perform an equivalent amount of work, whileproducing no other effect. (See, e.g., Sklar 1993, p. 21.)

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be extracted as work – which violates the Kelvin-Planck version of the second law of

thermodynamics.11

If we cannot find a way to block this result the generalized second law will also be

violated, for when the box is opened the amount of mass energy that is added to the black

hole is given by ,. If , can be made arbitrarily small, then the increase in the horizon

area *A will also be negligible. Thus, SBH, will remain effectively constant, while the

entropy associated with the thermal source can be lowered indefinitely.

Bekenstein’s answer to this challenge is to question the assumption that the box

could be lowered within an arbitrarily small distance of the event horizon. He points out

that the box will have to have a finite size, and argues that this size will place a limit on

the amount of thermal energy that it can contain. In an early article he argues that “the

box must be large enough for the wavelengths characteristic of radiation of some

temperature T to fit into it” (Bekenstein 1973, p. 2342). However, this argument assumes

the electromagnetic nature of the thermal energy, which appears to be overly restrictive.

Later (Bekenstein 1981) he proposed that there is a general bound on the ratio of entropy

to energy that can be enclosed in any container that will fit into a sphere of radius R. This

bound is given by

S / E < 2BR. (3.28)

If this bound holds then it will be impossible to violate the generalized second law by the

mechanism described above, for we will be unable to lower the box close enough to the

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event horizon to convert more thermal energy into work than is compensated for by the

increase in horizon area.

While it may be true, as Bekenstein argues, that this bound holds for matter fields

that exist in our universe, Unruh and Wald (1982) point out that it will be violated in any

possible world that contains a sufficient number of distinct, non-interacting, massless

types of matter. Thus (3.28), if true, seems to only be an accidental fact about the world

we live in, and does not provide a fundamental defense of the generalized second law.

Unruh and Wald’s alternative defense of the second law relies on the fact that

black holes are surrounded by a thermal atmosphere. Because the temperature felt by a

box slowly lowered to the event horizon will increase as it approaches the horizon, there

will also be a pressure gradient – thus the box will be subjected to a “buoyancy” force.

Just as the locally measured temperature of a black hole approaches infinity for fiducial

observers arbitrarily close to the horizon, so this buoyancy force will also become infinite

as the box is slowly lowered to the horizon. Therefore, if we wish to extract the maximal

amount of energy from the thermal radiation in the box, we should not try to lower the

box all the way to the horizon. Rather, we should open our trap door when the box

reaches the “floating point” associated with the buoyancy force due to the thermal

atmosphere of the hole. Calculations of this floating point (Unruh and Wald 1982) reveal

that the generalized second law cannot be violated by slowly lowering thermal radiation

close to a black hole and then releasing it.

There is no general proof of the generalized second law – and this fact will not be

surprising if we take the ordinary second law to be a merely phenomenological law that

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need not apply at the microlevel to all systems. However, the parallel between the

thermodynamic laws and the laws of black hole mechanics, the explicit derivation of the

temperature of black holes, and our ability to recover the generalized second laws from

scenarios that on the face of it would allow us to violate it, all lend support to

Bekenstein’s claim that the area of a black hole really is a measure of its entropy.

However, it is far from clear what this entropy is measuring. Typically we think of

entropy as a measure of the number of microstates that can fall under a certain macro-

description. Thus, a more or less obvious suggestion is that the entropy of the black hole

is a measure of the number of different ways that a black hole of a given mass, charge,

and angular momentum could be constructed. Once a certain bit of matter forms a black

hole the "microscopic" details of the matter are lost to us (since "black holes have no

hair"), and the Bekenstein entropy is a measure of how many details can be lost behind

the perfectly uniform surface of a black hole.

However, it is not immediately clear what precisely these micro-details might be,

nor how the Bekenstein entropy of a black hole could be such a huge, but still finite,

number. The order of magnitude of the black hole entropy for a solar mass black hole is

SBH/kB ~ 1077, as compared to a thermodynamic entropy of STD/kB ~ 1058. How is it that

there can be exp[1077] ways of making up a solar mass, zero charge, zero angular

momentum, black hole? Recently string theorists have claimed to find the answer to this

question – as we shall discuss in more detail in Chapter 6.

One further point worth highlighting is that the Bekenstein entropy increases as

the area of a black hole, and not as the volume as we would expect from any field

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theoretic, or particle theoretic, account of the microstates. This fact ties in with more

recent suggestions regarding a bound on the number of fundamental degrees of freedom

that can be contained in any region. This proposed bound goes by the name of the

“holographic principle”, and requires that the total entropy within any region of area

A must obey the following inequality:

S # A/4.

This bound would eliminate Bekenstein’s proposed bound, but, as we shall see in

Chapter 6, it faces objections similar to those that (3.28) faces.

12 Although this is the typical formulation of the significance of unitary evolution, it isworth noting that it is not strictly true. There is no set of measurements that we couldperform, even in principle, to discover the complete original quantum state of the system. We would need an ensemble of identically prepared systems (that is, a large number ofidentical encyclopedias destroyed in exactly the same way) first to establish thefrequencies of outcomes for one set of measurements, and second to allow us to measureobservables complementary to those measured by our original experiments.

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CHAPTER 4

THE INFORMATION LOSS PARADOX

We turn now to Hawking’s argument for the nonunitary evolution of systems that

include evaporating black holes, and some of the responses to his paradox. Popular

descriptions of the Hawking information loss paradox (e.g., Susskind 1997) describe

situations in which the information contained, for example, in an encyclopedia, a copy of

a mathematical formula, or a recipe for a French dish, is lost down a black hole. Had

these objects been blown up, or burned, the information they contained could still, in

principle, be recovered from the radiation given off and the ashes left behind.12 If the

objects were to fall behind the event horizon of a black hole, on the other hand, and if that

black hole were then to give off thermal Hawking radiation and eventually disappear,

then this information could never, even in principle, be recovered. The popular account

then reports that our physics (specifically quantum mechanics) does not allow for this sort

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of information loss at a fundamental level. Therefore evaporating black holes spell

trouble for physics as we now understand it.

As we shall discuss in Section 4.1, the formal argument for this breakdown does

not explicitly refer to information, but instead argues that because there will be

correlations between the inside and the outside of a black hole, the disappearance of a

black hole will necessarily leave the universe in a mixed state, even if it began in a pure

state. Such evolution is necessarily nonunitary and thus cannot be described by standard

quantum theory. However, there is a fair amount of debate over just how “non-standard”

a quantum theory must be to accommodate these transitions.

One of the more substantial objections against Hawking’s proposed account of

nonunitary evolution argues that such evolutions will lead to a violation of energy

conservation or locality. Advocates of Hawking’s conclusion, such as Unruh and Wald,

respond by claiming, first, that any such violations may be restricted to a realm beyond

the reach of our experiments and, second, that the arguments for the inevitability of such

violations are based on an inappropriate model of black hole evaporation. We shall

consider this debate in Section 4.2 after laying out Hawking’s model of pure to mixed

state transitions in more detail.

In Section 4.3 we discuss two (overly) simple responses to the information loss

argument: the bleaching scenario and the cloning scenario. Although neither view has

any defenders, because each faces decisive and well-known objections, they are useful

tools for explaining and evaluating more substantial responses. One such response, the

topic of Section 4.4, is the black hole remnant scenario, which attempts to save unitary

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evolution by allowing violations of microcausality at Planckian energies and distances.

Although this view has been abandoned by its former advocates, it still offers useful

insights into the nature of the debate over evaporating black holes.

Section 4.5 focuses on the question of how the various positions in the debate

over information loss in black holes should be understood and evaluated. We shall see

that the real debate centers on where and how our effective theories break down, and this

highlights the importance of considering the domains of validity of our effective theories

in our interpretive projects. Further, we require an understanding of the character of the

debate if we are to understand and evaluate black hole complementarity, which we

address in the next chapter.

4.1: Pure to Mixed State Argument

The foundation of the information-loss paradox is Hawking's discovery, discussed

above in Section 3.5, that black holes give off thermal radiation. This effect was derived

using QFT in curved spacetime, specifically the background spacetime of a

Schwarzschild black hole. On the face of it, our derivation has the seemingly implausible

consequence that a black hole will continually give off radiation at a temperature

inversely proportional to its mass for all future time, which would be a rather spectacular

violation of energy conservation. The straightforward response to this worry points out

that QFT in curved spacetime is inherently limited in that it includes the effect of

spacetime on our quantum matter fields, but neglects the effect of matter distributions on

13 See Appendix A for an explanation of Penrose diagrams.

112

the spacetime itself. We expect that as a black hole gives off energy through Hawking

radiation it will lose an equivalent amount of mass, thus balancing the energy books.

This expectation is supported, to a degree, by semi-classical quantum gravity.

This theory offers a classical account of spacetime – i.e., the spacetime is described by

classical general relativity – while the matter in that spacetime is described by QFT. The

stress-energy term in the Einstein field equation (Equation 3.1) is given by the

expectation value of the stress-energy of the quantum fields; that is, the theory is captured

by the equation

R:< ! ½ g:<R = 8B + ÎT:<,R, (4.1)

where ÎT:< is the quantum stress-energy operator and the expectation value is taken in the

quantum-field state |R,. This is therefore a first-order description of the back-reaction of

the quantum field on the spacetime. While these equations are too difficult to solve in

four dimensions, solutions to the simpler two-dimensional case confirm our expectation

that as energy is radiated away from black hole, its mass will decrease. If nothing halts

this evaporation process, the black hole will eventually disappear completely.

Figure 4.1 is a Penrose conformal diagram representing the formation and

complete evaporation of a black hole.13 The shaded area represents the black hole, the

region that is not in the causal past of future null infinity I+. The causal curves that pass

through the black hole cannot be extended to I+ but instead are assumed to terminate in a

curvature singularity indicated by the jagged line. Although the existence of a black hole

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Figure 4.1: Spacetime with evaporating black hole.

implies that a spacetime is singular, we will see that many physicists question whether

black holes harbor curvature singularities in their centers.

Consider the three spacelike hypersurfaces, Eearly, E0, and Elate, on which we can

define our global state at a “time.” The hypersurface, E0, is composed of a section inside

the black hole, which we will call Ebh, and a section outside, labeled Eext. We assume that

our quantum field begins in a pure state, Dearly, at an early time, that is, on our

hypersurface Eearly. Quantum mechanics tells us that this state will then unitarily evolve

into another pure state, D0, on E0.

At this point in the argument we appeal to the validity of local QFT, according to

which D0 will be a state in ,bh q ,ext, where these two Hilbert spaces will be defined by

local field operators on Ebh and Eext respectively. We can then define the component

states Dbh and Dext by tracing D0 over the degrees of freedom associated with Eext and Ebh.

14 One might wonder whether the relationship that a region inside the black hole has witha region of Glate is importantly different from its relationship with a region of Gext. Although there are no causal paths connecting the black hole region with either of theseexternal regions, the absence of causal paths between the black hole and (some region of)Glate is due to the presence of topological features of the spacetime (namely thesingularity) which might be thought of as being importantly contingent in some way. (If asufficient quantity of matter had never gathered in a sufficiently small volume, therewould be no singularity.) On the other hand, one might think that the fact that a sectionof Gbh is spacelike related to any section of Gext rests in the fundamental conformal natureof the spacetime; and thus the lack of causal paths between these two regions might seem,in some sense, less “accidental” than the former case.

However, it would seem to be a mistake to attempt to distinguish betweendifferent “types” of spacelike separation based on whether or not the lack of causal curvesrests on the termination of curves in a singularity. While classical GR does indicate adegree of contingency associated with the topological structure of spacetime, the

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These component states, Dbh and Dext, will be pure states if and only if D0 is an

uncorrelated, factorizable state, that is, if and only if

D0 = Dbh q Dext.

However, we have every reason to think that there will be quantum correlations between

the field on Ebh and the field on Eext. Previously interacting particles will generally be

entangled, and given that there seems no mechanism to prevent components of such

entangled systems from falling into the black hole, we should expect a generic black hole

to be entangled with some exterior system. Further, the derivation of Hawking radiation

explicitly indicates correlations between the outgoing Hawking radiation and the interior

state of the black hole. Since we expect that D0 will be an entangled state, we conclude

that Dbh and Dext will be mixed states.

Microcausality implies that any observable that can be constructed from the local

field operators on Ebh, will commute with all local observables defined on the late time

slice Elate, because these two regions are spacelike related.14 In the framework of QFT,

conformal structure of the spacetime will likewise depend on the details of the matterinhabiting the spacetime. The possibility of forming closed timelike loops, for example,indicates that large scale metrical features of the spacetime can undermine apparentlywell-defined local conformal features of the spacetime.

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this implies that our late time state, Dlate, will be independent of the local field values on

Ebh, and that Dlate will unitarily evolve from Dext.

But now we recall that Dext is a mixed state, which implies that Dlate is also mixed

(since unitary evolution preserves the purity or mixedness of states). It therefore appears

that the formation and complete evaporation of a black hole cannot be described by the

unitary evolution of states in a Hilbert space. This standard formulation of quantum

mechanics describes all evolution as unitary transformations of state operators, but the

evolution described above cannot be unitary since it begins with a pure state, Dearly, and

ends with a mixed state, Dlate. This is Hawking’s “paradox.”

4.2: Superscattering Matrices and Nonunitary Evolution

Before reviewing the various proposals that have been suggested for rescuing

unitary evolution from Hawking's argument, it may be helpful to investigate the potential

implications of nonunitary evolution. Indeed, it is not entirely clear that Hawking's

argument deserves the title of a “paradox” – why should we not simply consider it a

straightforward argument for the existence of nonunitary transitions between pure states

and mixed states in spacetimes containing evaporating black holes? Where is the

paradoxical element in this?

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The first point typically emphasized in replies to Hawking's argument is that all

evolutions in current quantum theories (excluding the problem of measurement) are

unitary evolutions. Thus any theory that allowed such pure to mixed state evolutions

would have to change at least one of the fundamental features of the quantum physics that

has enjoyed such tremendous success through the last century. While this may give us

reason to pause before discarding the requirement for exclusively unitary evolutions, and

while it may give us strong motivation to pursue avenues that might preserve this

requirement, it certainly offers us no guarantee that a revision might not be called for.

After all, the difficulties of developing a theory of quantum gravity seem to indicate that

major revisions of some sort are required in our heretofore unchallenged physical

assumptions. Unitarity might simply have to be sacrificed to effect a marriage of QM and

GR.

Support for this view can arguably be found in the problematic nature of energy

conservation in GR. Because there is generally no invariant measure of gravitational

energy in GR, it is not clear that we will generally have a well-defined account of

conservation of energy, and given the connection between energy conservation and time

evolution this may cast some suspicion on the validity of unitary evolution in this context.

The fact that some spacetimes described by GR, including the spacetime of an

evaporating black hole depicted in Figure 4.1, contain no Cauchy surface may give us

further reason to think that we shouldn’t expect unitarity to hold in such situations.

Indeed, in Figure 4.1 it may seem simply obvious that the presence of the singularity

indicates that we should not expect such a spacetime to behave like a closed system in its

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evolution from Eearly to Elate. Recall that to say that a spacetime is singular is to say that it

is geodesically incomplete; that is, not all world-lines can be extended to future timelike

or null infinity. It is therefore natural to view a singularity as an “edge” of spacetime (see

Earman 1995 for a discussion of this topic). As such, we might find it quite natural to

suppose that not all the information about our original state will be available at I+ – there

are no causal paths that could carry it there.

This response appeals to a rather realistic interpretation of the singularities in

solutions to the Einstein field equations: such singularities are viewed as features of the

spacetime that can potentially correspond to facts about our world, namely that our world

ends at the singularity. However, if we view these singularities as mere breakdowns in

the classical description offered by GR, rather than as physical features of spacetime, then

we will likely not be swayed by the above considerations of "losing" coherence via

termination of world lines. In this case, the singular nature of our classical equations

would offer no particular reason to believe that the evolution described by a full theory of

quantum gravity will be nonunitary.

We can bolster our intuition that such evolution should be unitary by pointing out

that a black hole could be formed by the collision of two sufficiently energetic particles.

At normal energies such collisions are described by an S-matrix that maps the incident

particles onto probabilities for observing outgoing particles as a result of the scattering.

As we raise the energies, eventually it will be likely that a black hole will form as a result

of the collision. Because it was formed by only two particles, this black hole will

generally be small, and thus very hot; it will evaporate away into a few outgoing particles

15 Although the originally proposed title for his information-loss paper, “The Breakdownof Physics in Gravitational Collapse,” might offer the opposite impression (cf. Page,1994).

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in a very short amount of time. We thus once again have a typical scattering event: a

couple of particles come together, interact, and then separate into easily described

outgoing particle states. It is then natural, especially for a particle physicist, to expect to

find an S-matrix describing the probabilities of the transitions between the ingoing and

outgoing particles, and the possibility that such an S-matrix seems not to exist might be

seen as a paradox.

Such intuitions and plausibility arguments, however, will hardly carry us very far.

The more rigorous objections to nonunitary evolution focus on questions of locality and

conservation of energy, and on the possibility of developing models of black hole

formation and evaporation that are manifestly unitary. We shall set aside this latter point

until Chapter 6 and focus here on the question of whether nonunitary evolution of

quantum states brings with it any particularly unpalatable features. Addressing this

question will require us to look a little more closely at the details of Hawking’s proposal.

In arguing that the evolution of quantum states on spacetimes containing

evaporating black holes cannot be unitary, Hawking did not mean to imply that such

evolutions cannot be described by some modified form of quantum theory.15 The

conclusion of his argument is that we should try to develop a theory that can

accommodate this pure-to-mixed state evolution. It is worth pausing briefly to consider

what such a claim would amount to. If such a theory were accurate we would still get

definite values on any particular run of any experiment: we create a black hole, let it

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evaporate, and measure the values of a set (ideally a complete set) of observables. If we

were considering only this particular system we would then describe it as being a pure

state represented by a common eigenvector of the observables measured. This would be

appropriate, for example, if we were using these measurements to select an ensemble of

systems on which we intended to perform further experiments. The ensemble so selected

would exhibit all the correlations of a pure state.

If we consider a series of such measurements on black hole formation and

evaporation, however, our measurement outcomes will generally vary. If we assign each

individual system its own pure state, then we will conclude that the final ensemble will be

a mixture of these states. If the evolution of the system can be described by standard

unitary evolution then by also measuring the outcomes of complementary observables we

can determine whether the original ensemble was in a pure state. If our measurements of

the amplitudes and the "complementary" phase information are precise enough then we

should be able to retrodict the original (pure) state of our ensemble.

Hawking's proposal is that even if we begin with an ensemble of systems all in the

same pure state, it will be impossible – in principle – to take the information from late-

time measurements and discover the original pure state of the ensemble. Another

implication is that given unitary evolution there will be some set of observables for which

we can predict the outcomes with certainty. This will not be true if the evolution of the

system is nonunitary. This is the basis for the claim that such evolution introduces an

extra level of indeterminism beyond the usual quantum indeterminism.

16 Note that the standard objection to nonunitary evolution offered by introductory textson QM is that such evolution does not preserve the norm of the state vector, and thereforeis inconsistent with conservation of probability. We avoid the force of this argument bydispensing with the representation of states by vectors and instead insisting on theemployment of density matrices to represent all states. This allows us to conserveprobability (d/dt Tr D = 0), while rejecting unitary evolution.

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In his 1976 article Hawking offered a suggestion of what such a theory might look

like. If we consider a quantum state D we can, in standard quantum mechanics, evolve

this state forward in time with a linear operator $, such that

DN = $D.

Normally this "superscattering operator" can be factorized into the usual scattering

operators discussed in Section 2.2,

$ = $ abc

d = S ac Sb

d †, (4.4)

which allows the evolution to be represented as in Equation (2.5): Dout = S Din S !1. If this

factorization is possible then S is a unitary operator, and the evolution represented by $

preserves the purity of states. However, we can consider nonunitary superscattering

operators that map pure states onto mixed states.16 These operators will still preserve the

unit trace of the density matrices, i.e., d/dt Tr D = 0, but will allow pure states to evolve

into mixed states, or more generally, the entropy of a state to increase. The entropy of a

quantum system, given by Tr(-D log D), is preserved by unitary evolution but not by a

more general superscattering matrix.

Hawking does not attempt to develop a full theory of quantum gravity in his

articles on this topic (1976, 1982), but instead assumes that we have some theory that will

allow us to calculate asymptotic Green’s functions (describing our asymptotic states) and

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asks what features we should expect those Green’s functions to have. The axioms he

suggests are essentially identical to the structure we laid out in Section 2.2 with the

important exception that he rejects the condition of asymptotic completeness. Recall that

this is the assumption that the creation operators defined in the asymptotic past, acting on

the vacuum state |0,, generate a basis for ,, and likewise for creation operators in the

asymptotic future. With this assumption in place, along with other axioms of standard

QFT, which Hawking accepts, one can demonstrate that the superscattering matrix can be

factorized as in (4.4), but if it is rejected then one can evolve pure states into mixed states

as desired.

Banks, Susskind, and Peskin (1984) published a reply to Hawking's article, which

argued that nonunitary evolution of the sort envisioned by Hawking would lead to

unacceptable violations of locality or energy conservation. These arguments were seen by

many physicists to raise serious difficulties for nonunitary accounts, and were often taken

as justification for more radical maneuvers for safeguarding unitary evolution. Unruh and

Wald have since published a paper (1995) arguing that the consequences of pure to mixed

state evolution are not as dire as the high-energy physicists imply. Although they do not

question the results of Banks et al., they do take issue with the scope of the arguments

and with the interpretation of the degree of energy conservation or nonlocality that

nonunitary evolution implies. Indeed, one gets the impression that the true target of their

objections is less the paper of Banks et al., and more the interpretation and weight that

others have since placed on the paper.

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Banks et al. begin their treatment of Hawking's position by supposing that the

dynamics of the system are Markovian, or local in time, meaning that the state only

depends on the previous instant in time, allowing us to write

$(t0, t2) = $(t1, t2)$(t0, t1).

Banks et al. offer a general account of the features of $ required for the preservation of

the trace of state operators, but we can consider a slightly more specific description of the

dynamics adopted by Unruh and Wald. We start with the Hamiltonian, H$ , of a standard

quantum field theory, but rather than the usual unitary evolution, dD/dt = !i[H$ , D], we

introduce the modified evolution law:

dD/dt = !i[H$ , D] ! Ei 8i(Q$ i D + D Q$ i - 2 Q$ i D Q$ i). (4.5)

Here the 8i are positive coefficients and the Q$ i are hermitian projection operators. This

evolution law demonstrably preserves the unit trace of the density matrix, i.e., Tr(dD/dt) =

0. It also allows the increase of the entropy of a state, i.e., d/dt [!Tr(D log D)] $ 0, as the

Q$ i introduce a mixing in the state by diminishing the off-diagonal terms in D.

The question is whether this evolution is compatible with locality and energy-

momentum conservation. Continuing to follow Unruh and Wald’s version of the

argument, we enforce a degree of locality by taking the Q$ i to project onto a subspace of

the Hilbert space that is acted on by a sufficiently local observable Âi associated with

region A. If we consider another local observable B$ i that is associated with region B

spacelike related to A, then microcausality will require that Âi and B$ i commute, and

therefore Q$ i will also commute with B$ i. This implies that the expectation values of B$ i

under the evolution (4.5) will be identical to the expectation values yielded by the

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standard unitary evolution of the state D. If our nonunitary evolution were not local in

this way – i.e., if Q$ i did not commute with B$ i – then it would apparently be possible to

signal over spacelike distances by inducing coherence loss (by forming a black hole and

allowing it to evaporate, for example) and observing the consequent change in the

outcomes of our measurements at B.

However, Banks et al. argue that such evolution is not compatible with energy

conservation. Let us consider the operator H$N (not necessarily equal to the above H$ ) that

will represent the energy of the system. If this energy is to be conserved under the

evolution given in (4.5), then any state operator that is a function only of H$N will have to

remain unchanged by this evolution. This is only possible if H$N commutes with all the Q$ i.

However, the local nature our projection operators Q$ i implies that they will not commute

with our momentum and energy operators. Thus it seems that energy conservation will

generally be violated by this nonunitary evolution.

Unruh and Wald do not dispute this claim, but argue that these violations can be

restricted to states that are inaccessible to laboratory experiments. We pick out the

subspace ,L of our Hilbert space corresponding to states we can access in our laboratory,

guaranteeing that this subspace is mapped onto itself by the unitary evolution governed by

H$ and that our regions Ai are sufficiently small that the states in ,L when restricted to any

Ai are essentially equivalent to some typical state such as the vacuum |0,. We can now

choose our 8i in Equation (4.5) such that the loss of coherence in D will be as rapid as we

like, while also choosing our Q$ i to guarantee that 8i||Q$ i|0,||2 is as small as we like. This is

accomplished by taking Q$ i to be a projection operator onto eigenvectors of Âi with very

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large eigenvalues, which will imply that the expectation value of Q$ i for the vacuum state

is negligible. With such a choice of 8i and Q$ i the evolution described in (4.5) will be

indistinguishable from standard unitary evolution for all observables that are accessible in

our laboratories, despite the fact that we have pure states evolving into mixtures.

The details of these models are not disputed by either party to the debate; indeed,

Unruh and Wald’s model is only a less general version of the model offered by Banks et

al. – who themselves point out that it is in principle possible to restrict the violations of

energy-momentum in such a way that they would be unobservably small. Unruh and

Wald’s article seems to be aimed primarily at those physicists who viewed the argument

of Banks et al. as a demonstration that nonunitary evolution would necessarily involve

either unacceptable violations of energy conservation or of locality. Against this common

sentiment they want to argue that nonunitary evolution can respect these principles, at

least at laboratory scales. However, the significance of their argument remains

questionable.

First, the target of the article by Banks et al. is Hawking’s (1982) suggestion that

the loss of coherence due to (potential) black hole formation and evaporation might have

a detectable influence on some scattering events. Banks et al. argue that this would imply

violations of energy conservation. Unruh and Wald’s response does not undermine this

conclusion, for their model restricts violations of energy conservation to inaccessible

states only by restricting all effects of the nonunitary evolution to these states.

Second, the general point seems to stand that nonunitary evolution implies

violations of locality or energy conservation; the only qualification that Unruh and Wald

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offer is that it is possible that our laboratory physics will not be exposed to this evolution

or its ill effects. Thus if we find it distasteful to assume that the formation and

evaporation of black holes will violate either locality or energy conservation – not,

primarily, because we have experimental evidence for the truth of these features, but

because we expect them to hold as fundamental principles – then we seem to have

reasons to expect (or hope) that such evolution will be unitary.

Third, Unruh and Wald have only shown us that there is a choice of parameters

and operators for which the effects of nonunitary evolution will not leak into laboratory

physics, but they have given us no reason to think that it is plausible to believe that

nonunitary dynamics would be restricted in this way. That is, we have no principled, or

natural, reason to believe that nature would restrict the violations of locality and energy

conservation to inaccessible states; we may be little comforted if our only protection from

such effects rests on ad hoc gerrymandering.

Responses to some of these worries are offered by both Hawking and by Unruh

and Wald; however the character of these replies is preliminary, and neither has been

developed into a viable model of unproblematic nonunitary evolution. Hawking claims

that the evolution described by $ should conserve energy asymptotically but not locally;

however, it is not clear that "merely" local violations of energy conservation are any

better than a denial of energy conservation simpliciter. Unruh and Wald argue that we

should expect the evolution describing an evaporating black hole to be non-Markovian,

but they have no viable model to offer. Their example, borrowed from Giddings (1995),

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does not allow for reproducible loss of coherence and is therefore not a realistic model of

black hole formation and evaporation.

In conclusion, while we have not found grounds to rule out decisively the

possibility that quantum gravity allows the nonunitary evolution of states, we have reason

to be suspicious of such a possibility as it generally seems to allow for violations of

locality or energy conservation. The question of whether these violations could be

contained in such a way to protect the validity of our very successful unitary, local

quantum theories remains open. Such containment is demonstrably possible in principle,

if put in by hand, but it has not been shown to follow in any plausible way from features

that one would expect from such a theory.

4.3: Bleaching, Cloning, and Information

Overviews of attempts to escape Hawking’s information loss argument almost

universally consider two proposals that have won no adherents: the bleaching scenario

and the cloning (or xeroxing) of quantum information. Although these views face

decisive objections, they play a useful pedagogical role in clarifying the difficulties we

face in resolving the information loss paradox. If, following the popular gloss on the

paradox, we view the heart of the problem to be the disappearance of information behind

a black-hole horizon from whence it cannot return, we might wonder whether the

information in question couldn’t somehow be left behind at the horizon, allowing it to

return to the external universe and saving physics as we know it. This is the bleaching

scenario. Another option that might occur to us is the possibility of duplicating the

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precious information before it disappears behind the horizon. This cloning process might

then allow the information to escape as well as fall into the black hole.

As we shall see momentarily, neither of these suggestions holds up under

investigation. However, we shall be interested in these proposals for one further reason:

they form the Scylla and Charybdis between which black hole complementarity tries to

navigate. Recall that, according to BHC, an outside observer will see any information

falling into a black hole burn up on the heated stretched horizon. This sounds precisely

like a description of the bleaching scenario. The infalling observer, however, will have a

very different view of things: she will notice nothing unusual around the event horizon

and will pass uneventfully into the interior of the black hole. Thus one might be tempted

to claim that objectively the information carried by this observer must have been

duplicated, as it is both inside and outside the black hole. To understand how and why

the black hole complementarians hope to avoid these positions we will need to consider

the details of these two proposals and the problems that they face.

We begin, then, with the quantum bleaching scenario. Hawking’s argument for

nonunitary evolution was driven by the claim that the state of the universe outside a black

hole is entangled with the state inside the black hole. If our joint state were not

entangled, but factorizable, then the state of the universe after the disappearance of the

black hole could remain pure. Could we then suppose that there is some mechanism at

the event horizon of the black hole that prevents quantum correlations from forming

across it, i.e., some mechanism that guarantees that the joint state of the interior and

exterior of the black hole will always be the factorizable pure state |R, = |Nbh, q |Next,

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(where for visual clarity we designate the joint system by R and the component systems

by N).

The flaw in this suggestion is that such a mechanism would require the

destruction of any object falling into a black hole. This conclusion follows simply from

the linearity of quantum evolution. Suppose that we have two non-orthogonal global

states |R1, and |R2,. Our mechanism enforcing the factorizability of interior and exterior

states will then imply that these two states can always be factorized as

|R1, = |N1bh, q |N1

ext,; |R2, = |N2bh, q |N2

ext,.

The fact that quantum evolution is linear, however, implies that a linear superposition of

these states |R1, + |R2, (ignoring normalization factors) will also be a solution of our

equations of motion. Using our above characterization of these states we have

|R1, + |R2, = (|N1bh, q |N1

ext,) + (|N2bh, q |N2

ext,).

However, our mechanism for enforcing the purity of our exterior state will have to

guarantee that this state is factorizable. This is only possible if | N1bh, = |N2

bh, or if

|N1ext, = |N2

ext,; since this argument applies for any |R, and any |N,, however, we conclude

that the purity of the late time state requires the same state either inside or outside the

black hole regardless of any other details of the universe, such as the past history of the

universe.

Our immediate experience tells us that the state of the exterior of black holes (e.g.,

of our daily lives) does change and is influenced by past events. Thus our only option for

enforcing the factorizability of the interior and exterior of black holes is to have the state

of a black hole’s interior always be the same. This, however, means that it will be

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impossible to enter a black hole without being destroyed, as the interior state will be the

same regardless of which observer tries to enter, or whether any observer or object tries to

enter it at all. We therefore conclude that any mechanism that will reliably keep the state

outside a black hole pure will have to prevent anything from successfully crossing the

event horizon, either by blocking its transit or by destroying it.

This possibility, however, is in conflict with the fact that the event horizon of a

black hole is a globally defined object that should be completely unremarkable locally: a

free-falling observer should notice nothing out of the ordinary in the vicinity of the event

horizon of a sufficiently large black hole, but her own destruction (or that of her

companions) is certainly something that she would notice. The argument therefore

concludes with the claim that there can be no such mechanism enforcing the purity of the

late-time state – at least for sufficiently large black holes that should be adequately

described by classical general relativity.

Our next suggested escape is that perhaps we could keep our late time state pure if

the infalling information were somehow duplicated at the event horizon. This is

supposed to allow the information both to fall into the black hole, and also to be returned

to the external universe. However, overviews of the information loss paradox quickly

point out that there can be no linear quantum transformation that will duplicate a general

quantum state. The argument goes as follows:

Suppose that we have a xeroxing (cloning) process, xY, that will duplicate any

quantum state, e.g., |A, or |B,:

|A, xY |A,q|A, |B, xY |B,q|B,

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If this process can xerox every state it will also have to duplicate a linear superposition of

|A, and |B,, e.g.,

(|A, + |B,) xY (|A, + |B,) q (|A, + |B,).

But this is incompatible with the xeroxing procedure being linear, because linearity

requires that

(|A, + |B,) xY |A,q|A, + |B,q|B,,

and this differs from the previous result by the absence of cross terms.

Note, however, that this argument does not prohibit all duplication of quantum

states; it only implies that there cannot be a single process that can duplicate every state,

or more specifically, that no single linear process can duplicate both a given state and a

state that is not orthogonal (or identical) to that given state. If I happen to know the state

of the system that I want to duplicate, there is nothing to prevent me from doing so;

indeed, our ability to measure quantum states typically relies on the possibility of such

amplification. All this is well known, and is mentioned in Wootters and Zurek's seminal

paper on quantum cloning (1982), but it is never emphasized in (and is sometimes

omitted from) discussions of information loss in black holes.

To explore the relevance of this qualification let us suppose that we are going to

form a black hole from matter in a known quantum state, say a number of particles all

prepared in definite spin states. I am then charged with finding some mechanism to

guarantee that the state of the universe at late times remains pure. Since I know the state

of the incoming particles, I can, if I wish, duplicate them and send one into the black hole

and one out to infinity. But notice that I have absolutely no motivation to do so, since by

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hypothesis each of these particles is in a pure state and thus is not correlated with

anything in the external universe. I can therefore allow the particles to form a black hole,

evaporate away, and disappear, without fear of losing the purity of the universe as a

whole.

Suppose, on the other hand, that we instead form the black hole from particles that

are all entangled with partner particles, e.g., suppose that each particle is a component of

a singlet spin pair. Then when the particles undergo gravitational collapse, leaving their

respective partners outside the forming black hole, loss of purity threatens. What options

do I, the quantum cloner, have open to me in my attempts to preserve the “information”

that is falling into the black hole?

First note that because I know the joint state of each of these particle pairs I can

duplicate the infalling particles by also duplicating its external partner. That is, I can

produce an indistinguishable set of entangled particles. But clearly this will do nothing to

prevent the loss of coherence once our original particles disappear down the evaporating

black hole. Our original particle pairs will still be entangled, which means that the

component states will be mixed and our late time global state cannot be pure. The only

other description available is the reduced density matrix that encodes the probabilities for

measurements performed on a single particle. I might then produces a mixture of

particles that would be described by the same density matrix, but again this will do

nothing to save the purity of the late time state. These are the only options open to me:

there is no other "state," no other "information" that belongs to the infalling particles.

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Our argument for information loss (nonunitary evolution) goes through when (and

only when) we have a correlated joint state and then we lose a component of this state.

This means that if we have a pure component state (which, therefore, is not entangled

with any other state) falling into a black hole, then we have no information loss,

regardless of whether or not we clone this system. On the other hand, if we have an

entangled joint state, then no amount of state duplicating will prevent the final state from

being mixed if one of the component systems disappears.

Glossing the problem of nonunitary evolution as a problem of losing

information is therefore misleading insofar as it implies that we have lost something that

can be captured by a physical description of the system that falls into the black hole. The

plausibility of the quantum cloning scenario as a response to Hawking’s paradox derives

from an illegitimate reification of descriptions of infalling matter. Once we see that our

ability or inability to duplicate systems is irrelevant to the question of whether the

evolution under consideration is unitary, we can see that the quantum cloning response is

a red herring.

On the other hand, there are legitimate respects in which the concept of

information enters into the debate, so the title “information loss paradox” is not a

complete misnomer. While a complete analysis of the concept of information in classical

and quantum theories goes well beyond the scope of this dissertation, a few points will be

useful for our discussion. The first is that information can fruitfully be seen as the inverse

of entropy (indeed, it is occasionally referred to as “negentropy”). If we classify states of

a system by energy, then the state of maximum entropy – the thermal state

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DT = exp[!H$ /T kB] – is the mixture that tells us the least about the details of a particular

system; that is, this state contains more possible micro-states than any other state

compatible with the energy-classification. In this sense, then, such a thermal state can be

regarded as carrying no information beyond its temperature T; it is, we might say,

maximally mixed.

However, mixed states can also be the result of considering sub-components of

entangled systems. For example, if we consider a pair of EPR particles, each individual

particle will be described by a mixed state (indeed a thermal state for certain EPR pairs),

but the overall joint state is obviously pure. If we consider a region of spacetime

containing only one of the particles, we would be inclined to assign it a nonzero entropy –

and an ensemble of such particles would behave as a thermal mixture. However, if we

reintroduce the partner particle, the entropy of the region will return to zero – and we

could (in principle) experimentally demonstrate the purity of our state. The moral here is

that subsystems that locally look like highly mixed states can nevertheless be part of a

global system that is not mixed, and another apparently mixed subsystem can allow us to

make manifest the purity of the overall state.

This is possible because of the quantum correlations that exist between entangled

states. In this sense it is reasonable to claim that one particle carries “information” about

the other, in that a measurement on one component of an EPR pair will reveal what the

result of the same measurement on the other component will be. If one of these particles

were to be lost, we would have lost our only means of predicting this outcome, and in this

sense information about the one particle would be lost when its partner was destroyed.

17 The following two sections are drawn from P. Bokulich (2001).

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The popular gloss on the information loss paradox has it that the information that

enters into the black hole cannot be reemitted in the outgoing radiation because Hawking

radiation is completely thermal. The more important question, however, is whether this

outgoing radiation might be correlated with the external universe in such a way that the

purity of the late-time state can be restored. As we shall see in the next section,

microcausality implies that this cannot happen.

A last point on this topic is that Hawking’s derivation purports to offer a

description of the entire field system on I+ and not just a description of some subsystem;

if his derivation can be trusted then the final state is a thermal mixed state, and the

evolution cannot be unitary. A defense of unitary evolution that wishes to accept

Hawking’s account as effectively accurate must therefore insist that the outgoing

radiation is not truly thermal, but actually harbors hidden correlations. The radiation will

look mixed to all practical measurements, but it is nonetheless actually pure.

4.4: Black Hole Remnants17

Defenders of a unitary description of the formation and evaporation of black holes

typically come from the camp of high-energy particle physicists, which includes string

theorists. Two types of models have been put forward by these defenders: remnant

scenarios, the topic of this section, and black hole complementarity, the topic of the next

two chapters. Both of these responses argue that some aspect of the above picture of

evaporating black holes will be excluded by a full theory of quantum gravity, and thus we

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can expect such a theory to retain unitary quantum mechanical evolution. The remnant

proposal, in particular, points to the fact that the semi-classical methods used to derive

the Hawking effect are clearly invalid when the black hole reaches the Planck mass. This

opens the possibility that quantum gravitational processes might safeguard, and perhaps

even return, the information contained in the black hole, thus allowing for the unitary

evolution of our global quantum state.

If Planckian physics were simply to shut down the Hawking radiation, then we

would be left with a very small black hole that would be quantum entangled with the

external state of the universe. We might call such an entity a “residual black hole” in

recognition of the fact that it retains the essentially spatiotemporal character of a black

hole. By contrast, we can consider a scenario that postulates the evolution of a Planck

sized black hole into some new object: a “black hole remnant.” Such a remnant is

typically described as a fundamental particle, of an essentially quantum gravitational

nature, that retains all information that falls into a black hole.

If the remnant, or the residual black hole, remains for all future time, then we call

it a stable remnant. However, there is also the possibility that Planckian physics will

allow the information stored in the remnant to return to the external universe. Once all

the correlations are passed off to field operators localized outside the remnant, thus

securing the purity of the external state, the remnant can safely pass out of existence.

These transient objects are referred to as long-lived remnants in recognition of the fact

that their finite lifetime will nonetheless have to be quite lengthy, given the potentially

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immense amount of information that they will have to return, and the meager Plank-sized

mass they have to return it with.

The amount of information that these remnants would have to store is the basis of

the most substantial physical objection to this proposal. There would appear to be no

bound on the number of internal degrees of freedom that a remnant would have to

possess. Suppose, for example, that we form a black hole from a large amount of matter

that is entangled with some system outside the black hole. We can then wait for the black

hole to shrink by giving off thermal radiation that will be uncorrelated with any external

system. We then feed in more correlated matter, let the hole shrink, and so on. There

seems to be no bound on the number of correlations (the amount of information) that the

black hole – and hence the remnant – would have to support. However, this seems to

imply that a remnant should have an indefinitely large number of internal states, while

only having Planckian mass. This in turn implies that such remnants should be pair

produced without bound in background electromagnetic or gravitational fields, which is

clearly false. (See, for example, Giddings 1995b, for more details of this argument.)

These physical challenges facing the proposal will not be our primary concern

here, however; instead we shall consider a charge leveled by Belot, Earman, and Ruetsche

(hereafter, BER) that the remnant scenario is an incoherent response to the Hawking

information loss argument. BER’s 1999 paper begins by offering a derivation of pure to

mixed state evolution, and then offers a taxonomy of proposed solutions to the paradox

based on which premise of the derivation is denied by a particular solution. Given their

aim of offering a taxonomy of responses to Hawking's Paradox based on how the

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response rejects a premise of their argument, what is BER’s evaluation of the remnant

proposal? They claim that the proposal faces a “fundamental difficulty,” which they pose

in the following dilemma:

Either remnants are remnants--that is, of black holes--in which case they do notprovide for a satisfying resolution of the Hawking paradox, or they are notremnants--at least, not of black holes--in which case they can do nothing toaddress the problem of black hole evaporation. (BER 1999, p. 216)

The first horn of the dilemma, which will be discussed more fully below, points out that

the derivation of pure to mixed state evolution rests solely on the commutation condition

(microcausality), and the claim that there are quantum correlations between the interior

and exterior of the black hole (BER’s “correlation condition”), where this black hole

evaporates as indicated in Figure 4.1. The mere postulation of information retaining

remnants, according to BER, does not appear to block any premise used in their

derivation, and therefore this response is not “satisfying.” On the other hand, as we shall

see below, one could see the remnant proposal as denying the existence of black holes.

But this move, according to BER, merely changes the subject and does not address the

concern at issue. Thus either remnants should be grouped with inadequate, confused

responses to the paradox, or they should be seen as (apparently uninteresting) denials of

the original assumption that black holes exist. I will argue below that the solution offered

by remnants is both satisfying and completely relevant to the question of evaporating

black holes, regardless of which horn of BER's dilemma we decide to face.

First, however, let us consider the case of residual black holes, that is, Planck

scale black holes that no longer Hawking radiate. It is not clear whether BER take their

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Figure 4.2: Banks’ residual black hole.

criticisms to apply to this remnant scenario as well, or whether they consider it to be

outside the scope of their discussion, as it denies the premise of complete evaporation.

This is nevertheless a prominent remnant scenario that clearly deserves our attention.

Such a scenario has been advocated by Banks (1995), who argues that Planck-scale

physics might shut down the Hawking radiation and prevent the formation of a singularity

in the center of the black hole. As we can see in Figure 4.2 (copied from Banks 1995),

appropriate late-time slices will include all the matter that fell into the black hole, and

thus our global state can remain pure. Although at any time the partial state associated

with the region that is accessible to an external observer will be mixed, the evolution of

the total state will be unitary, and thus we have a response to Hawking's argument. This

proposal violates BER’s commutation condition because there will be no time slice, Glate,

whose observables will all commute with the black hole region. This is a simple

consequence of the fact that the black hole continues to exist for all time. While there

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Figure 4.3: BER’s remnant scenarios.

may be physical reasons to reject this proposal – based, for example, on the

aforementioned worries about black hole pair production – it is nonetheless a

conceptually adequate response to Hawking’s argument.

Banks’ proposal explicitly retains the spatiotemporal nature of the end product of

black hole evaporation, which is therefore a residual black hole or a “remnant black hole”

as BER also refer to it. The main target of their argument, however, is the black hole

remnant proposal, which claims that Planck scale processes will replace the black hole

with a new sort of fundamental particle, namely a black hole remnant.

A way of picturing remnants which does not involve a residual black hole is givenin [Figure 4.3]. The spacetime in question still has the event horizon structureconstitutive of a black hole, so while the remnant (the ??? of [Figure 4.3]) is not aremnant black hole, it is [a] remnant of a black hole, and so confronts thedilemma's first horn. In this situation one can proclaim as loudly as one wantsthat information is stored in the remnant. Be that as it may, observables in thealgebra associated with post-evaporation slice G10 . . . (stable remnant) or G11 . . .(long-lived remnant) ought to commute with observables associated with theblack hole interior. (BER 1999, p. 216)

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BER consider one possibility for fleshing out the remnant story. The remnant (the ??? of

their diagrams) could be a placeholder for a set of boundary conditions imposed at the

singularity that would preserve unitarity.

But until remnant enthusiasts produce the new physics that incorporates theboundary conditions in a natural way, the present proposal ‘solves’ theinformation loss paradox only by inserting the missing information by hand, and‘remnant’ is just a name that does nothing to justify the sleight of hand. (BER1999, p. 218)

Leaving aside the question of whether this is a fair appraisal of the attempt to save

unitarity through the imposition of boundary conditions, we should recognize that

remnant theorists have other resources available.

The most common direction for the remnant proposal to take, and I think the most

promising, is to deny that there is a true singularity in the center of black holes. (Whether

these objects then deserve the title ‘black hole’ is a question we shall address in a

moment.) Indeed, it seems that this how Giddings (1995a) intended the reader to

interpret his figure that served as the model for BER's diagrams reproduced above.

While Giddings does not explicitly label the remnant in Figure 4.4, it seems clear that he

intends the remnant, and not a curvature singularity, to be the terminus of all causal paths

in the black hole. Why should this be a remnant rather than a singularity? Consider the

claim that the remnant theorist is making: A large amount of matter undergoes

gravitational collapse. This matter steadily loses mass through Hawking radiation, but it

remains entangled with the external world. Eventually the black hole shrinks to Planck

size and evolves from a black hole into a Planckian remnant. In the absence of evidence

for a contrary interpretation, it would seem that we should take the claim that the

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Figure 4.4: Giddings’ black hole remnant.

information of the infalling matter is transferred to the remnant to imply that the remnant

is in the causal future of this matter.

While Figure 4.4 represents a long-lived remnant, we can also correct BER’s

representation of a stable remnant by replacing the spacelike singularity with a remnant,

as indicated in Figure 4.5. Both the long-lived and the stable variants of this remnant

proposal escape BER’s argument by denying their commutation condition. The late-time

slice, Glate, includes the remnant, which in turn is in the causal future of all observables

localized inside the black hole. Thus these observables will generally not commute with

all observables on Glate. In the case of stable remnants only the observables associated

with the remnant itself will fail to commute with observables localized inside the black

hole. (Here we simply assume that it is legitimate to refer to the remnant degrees of

freedom as ‘observables,’ even though they cannot even in principle be measured by a

macroscopic observer.) If we have a long-lived remnant, on the other hand, field

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Figure 4.5: Adjusted diagram of remnant scenario.

observables in the causal future of the remnant will also fail to commute with the black

hole observables.

We now face the question of whether this denial of the commutation condition

entails a violation of microcausality. If our spacetime were adequately represented by

Figure 4.1, then the black hole region and Glate would clearly be spacelike related, because

there is no causal curve connecting these two regions. However, now that we have

replaced the singularity with a quantum gravitational remnant, what can we say about the

existence, or lack of existence, of causal paths out of the black hole?

We seem to have two options. The first is to agree that there are no causal (i.e.,

luminal or subluminal) curves out of the black hole, but claim that in or around the

remnant spacelike related observables will fail to commute. Our second option is to deny

that all of Glate is spacelike related to the black hole region, either because there are causal

paths through the remnant to Glate, or because the underlying quantum nature of the

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spacetime does not allow the unambiguous specification of whether these regions are

spacelike or timelike related. While remnant theorists often do not distinguish between

these two ways of fleshing out their position, their rejection of the commutation condition

at the Planck scale is typically quite explicit. For example, while discussing his long-

lived remnant scenario (cf. Figure 4.4 above) Giddings writes:

Another possibility is that the information is radiated after the black hole reachesM - mpl and the semiclassical approximation fails. Here ordinary causality nolonger applies to the interior of the black hole, and it's quite plausible that theinformation escapes. (Giddings 1995a, p. 551)

But is this failure of “ordinary causality” due to superluminal interactions across

Planckian distances, or a breakdown in the causal structure of spacetime? Or is this

distinction somehow ill posed in a truly quantum theory of spacetime?

In support of reading the proposal as postulating superluminal information

transfer, we can point out that remnant theorists typically claim that remnants are

fundamental Planck-sized particles. The claim that they are fundamental presumably

implies that they have no internal degrees of freedom of a spatial nature, i.e, they will

have no spatially related dynamically distinguishable parts. But if they have a finite

(Planckian) size, we can apparently no longer insist that all spacelike related observables

commute. If a portion of an extended remnant is in the causal future of a black hole, and

another portion of the remnant is in the causal past of a region of Glate, then an observable

associated with the black hole will generally not commute with an observable defined on

that region of Glate. The late-time observable need only be in the causal future of

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some portion of the spatially extended remnant, not necessarily the portion that lies inside

the lightcone of the black hole observable.

If we take the second alternative and argue that the black hole region is not

(definitely) spacelike related to Glate, then we face both the question of the accuracy of

Figure 5, and the second horn of BER’s dilemma; for it is not clear that such a suggestion

is compatible with the claim (depicted in the diagrams of the remnant spacetimes) that

these regions are not in the causal past of I+, which is the definition of a black hole. One

response to this challenge is to claim that the diagram is limited: the classical regions

have been accurately portrayed, but not all of spacetime is amenable to this classical

description. We misunderstand the proposal if we argue that because there are two

regions of spacetime that would be spacelike related if the spacetime were completely

classical, therefore operators associated with these regions “ought to commute.” Unlike

the stable residual black hole scenario postulated by Banks, an essential part of this

remnant scenario is a specification of regions where our classical approximations break

down, and it would be a mistake to insist on the commutation condition in this regime.

However, we still have not fully confronted the second horn of BER's dilemma,

for some versions of the remnant scenario imply that, strictly speaking, there will be no

points that fail to be in the causal past of I+. BER criticize one such scenario, due to

Giddings, on precisely these grounds: it denies the existence of a global event horizon,

and thus denies the existence of a black hole. The spacetime of Giddings' massive

remnant proposal is sketched in Figure 4.6 (reproduced from Giddings 1992, Figure 3).

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Figure 4.6: Giddings’ Massive Remnant Proposal.

Here the collapsing matter does not form a singularity, but rather a Planckian “core” that

superluminally expands past the horizon, thus allowing the information to return to the

external universe. BER question whether it is appropriate to refer to this proposal as a

solution of the problem of information loss in black holes, since there is no true black

hole in Figure 4.6.

One disturbing feature of this proposal is that the core expands at a superluminalrate. Worse, [Figure 4.6] incorporates no singularity of gravitational collapse. . . .Because the spacetime lacks a genuine black hole, the surface labeled “horizon” in[Figure 4.6] must be an apparent horizon, an object locally delineated, rather thana true event horizon. . . . And because the relevant event horizon structure ismissing, the theorems that underwrite Hawking radiation do not apply. In short,the labels of “black hole” and “black hole evaporation” strike us as misnomerswhen applied to [Figure 4.6]. . . . It is less a solution to the information lossparadox than a sweeping denial of the problem. But perhaps that is his point. (BER 1999, p. 219)

While I cannot fully address all of the objections BER raise here, a number of points

should be made on behalf of remnant proposals that deny the existence of singularities.

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The question of how we should use the term “black hole” is, of course, not the significant

issue here. We can, if we like, identify black holes by the formation of apparent horizons.

This was the concept that Wheeler first applied the term to in 1967, five years before

Hawking introduced his account of global event horizons. Alternatively, we could, with

very little violence to the term, identify a black hole with the region from which all causal

curves necessarily terminate in a Planckian remnant. We might call the border of this

region a “remnant horizon” if we wanted to distinguish it from an apparent horizon. Note

that nothing could escape this region without passing through the Planckian remnant, a

prospect not even a fundamental particle is likely to survive. But nothing of consequence

hangs on this terminological issue.

The substantial question is whether apparent horizons or remnant horizons will be

adequate for the derivation of Hawking radiation. Presumably it is this worry that

prompts BER to charge Giddings with offering a “sweeping denial of the problem.” If he

were denying the force of the argument for Hawking radiation, there might be something

to this charge (although I would still worry about the distinction between solving a

problem and denying that there is a problem). But, while Hawking's derivation of black

hole radiation rests on the presence of a global event horizon (and hence the existence of

a singularity), one can consistently accept the predictions of Hawking’s semiclassical

model while denying that the spacetime in question is singular. We shall see in Chapter 5

that one can produce explicit (if simplistic) models of black holes that evolve unitarily but

still produce Hawking-like radiation, demonstrating that nonsingular spacetimes can

reproduce the effective descriptions offered by Hawking.

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4.5: The Limits of Local QFT

Granting that Hawking-like radiation is possible in the absence of true black

holes, we still might wonder why a remnant theorist would go through the effort of trying

to recover Hawking’s prediction of black hole evaporation. One typically does not see

explicit arguments for the claim that large black holes will give off Hawking radiation

even if quantum gravitational processes prevent the formation of a true singularity and

global event horizon. This is because all the participants in the debate generally agree

that Hawking’s prediction will be effectively accurate for large black holes. The debate

centers around how accurate this effective description is likely to be (e.g., whether the

radiation will be truly thermal as predicted by Hawking), and what happens when the

effective description breaks down, for example, when the black hole shrinks to Planck

size. The remnant theorist agrees with Hawking that the semi-classical description ought

to be highly accurate on time-slices that stay well away from regions of strong

gravitational curvature. This implies that a large black hole should behave as if it were in

the process of forming a true curvature singularity throughout most of its lifetime.

Therefore it should be giving of thermal Hawking radiation – and any Planckian

mechanism that might prevent the formation of a true singularity would seem to be

unable to shut down this early time radiation.

The conviction that Hawking’s prediction should be accurate for early times,

along with the rejection of the claim that black holes completely evaporate away

destroying quantum coherence, led remnant proponents to look for an explicit model,

however idealized, of black holes evolving into remnants. The central project that

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occupied these theorists in the early 1990s was trying to find a simplified (usually two-

dimensional) unitary model of a black hole giving off radiation and shrinking, while still

evolving unitarily. Although a plausible quantum gravitational model for a black hole

remnant was never found, it should be clear that it would be illegitimate to dismiss such a

project as not addressing the question of information loss in black holes.

There are serious problems facing the remnant proposal. Indeed, I know of no

physicist who still defends this position. Giddings and Banks, for example, have both

abandoned the remnant view in favor of something like black hole complementarity

(private communication). However, these problems do not lie in a failure to address

Hawking’s argument adequately – if the scenario were plausible, it would resolve the

information loss paradox – but rather in physical predictions that are difficult to reconcile

with other aspects of physics (e.g., low mass objects having an exceedingly large number

of internal degrees of freedom), and in an apparent inconsistency between the predicted

behavior of remnants and the quantum gravitational theory that currently seems most

promising to the majority of high energy physicists, namely string theory.

Let us conclude this chapter by extracting some lessons from the foregoing

discussion concerning how we should confront scenarios in which we expect our current

theories to break down, as in the case of evaporating black holes. In these situations we

can often do considerably more than simply guess at how a system will behave and report

that our current theories are not completely trustworthy. We often have the resources to

develop some picture of where and how the theories will break down. In the case of

black hole evaporation, for example, we can appeal to the fact that local QFT begins to

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run into problematic infinities when we consider very short distances, and the fact that

interactions at Planckian energies should themselves create black holes, to argue that our

current theories are inadequate in such a regime. As the remnant theorists emphasize,

however, these same arguments seem to indicate that the semi-classical approximation

should be adequate when these energies are absent.

BER recognize that QFT and GR are only approximately true, but they claim that

“we cannot know how good the approximation is, or even what ‘approximately’ means,

until we know how to combine QM and GR in one theory” (BER 1999, p. 221). While

there is a good deal of truth in this statement, it neglects the fact that we sometimes can

establish some reasonable boundary lines for the validity of our existing theories.

Further, we can hope to glimpse some features of a full theory of quantum gravity by

working to establish the proper domain of the low energy theories we expect it to reduce

to. For this reason we should see the ??? in Figure 4.5 not merely as a placeholder for

some future physics, but also (and perhaps more substantially) as a proposal for the

proper border of our semi-classical description of the situation. Of course, remnant

theorists aimed at more than the demarcation of our current theories, but the question of

when we should expect the effects of Planckian physics to become manifest is an

important and controversial question; indeed, it is the central question under debate in the

controversy we are considering here.

Hawking’s argument for information loss, no less than the remnant proposal,

relies on assumptions about how and where GR and QFT will break down. The claim

that a black hole will continue to radiate when it has reached Planck size can only be

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justified by assumptions about how a full theory of quantum gravity is likely to behave.

Even the claim that a singularity and a global event horizon will form from the

gravitational collapse of a sufficiently massive body requires us to make assumptions that

go well beyond our semi-classical models, for the singularity theorems rest on energy

conditions that are generally violated by quantum fields in curved spacetimes. Indeed, it

is the violation of these energy conditions that allows us to escape Hawking’s area

increase theorem and claim that black holes shrink as a result of giving off radiation.

The argument for black hole complementarity also focuses on the question of

where and how the descriptions offered by semiclassical quantum gravity will break

down; advocates of this position, however, argue that Planckian physics will not be

restricted to the center of a black hole, but will become apparent at the event horizon as

well. To say that such processes will become apparent at the event horizon need not

commit one to the claim that there will necessarily be some locally observable processes

in that region that can only be described using a full theory of quantum gravity. The

details of precisely how the description offered by the low-energy effective theory breaks

down can be considerably more subtle, as we shall see in the specific proposals we

consider in the next chapter.

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CHAPTER 5

BLACK HOLE COMPLEMENTARITY

In this chapter we reach the core of the dissertation: here we explicate and

evaluate Black Hole Complementarity (BHC) as a solution to Hawking’s paradox and as

an example of Bohr’s complementarity. The three principle groups defending this view

are assembled around ’t Hooft, Susskind, and the Verlinde twins; we shall consider each

of their accounts in turn.

After laying out the essential details of ’t Hooft’s position in Section 5.1, we shall

turn in Section 5.2 to the philosophical grounding for his attempt to avoid the force of

Hawking’s paradox. The challenge that ’t Hooft and the other advocates of BHC face is

that of establishing the incompatibility of the measurements performed inside and outside

a black hole – because they are committed to the claim that these measurements are

complementary to one another. As we shall see, ’t Hooft’s primary justification for this

claim rests on an illegitimate appeal to measurement collapse, and his recent attempts to

develop a deterministic subquantum theory also do not seem to open an avenue for

complementarity to help resolve the information loss paradox.

The position of Susskind and his coauthors is considered in Section 5.3, while

Section 5.4 focuses on the conceptual justification offered for the position and the

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question of whether complementarity plays any role in rescuing unitary evolution. We

shall see that while we cannot rule out the possibility that the relevant observables are

complementary in Bohr’s sense, the arguments presented for this conclusion miss their

mark because they conflate complementarity with operationalism.

The most promising formulation of BHC is offered by Kiem, Verlinde, and

Verlinde, who postulate the needed incompatibility – though they offer little by way of

justification for this postulate. Nonetheless, their account helps to clarify the structure

that a theory would require if it were to exhibit the behavior necessary for BHC to resolve

Hawking’s paradox. In Chapter 6 we shall consider the suggestion that certain models of

quantum gravity may exhibit this behavior, and thus give us reason to believe that the

account offered by black hole complementarity may be accurate, despite the fact that the

previous arguments offered in support of the relationship are uncompelling.

5.1: ’t Hooft’s S-Matrix Ansatz

The first real champion of information retrieval from black holes – that is, of the

existence of a unitary S-matrix describing the formation and evaporation of black holes –

was Gerard ’t Hooft. He argued that we should be suspicious of Hawking’s claim that the

evaporation of a black hole entails a breakdown of unitary quantum mechanics, both

because the derivation of the late-time thermal state ignores the back-reaction of the

quantum state on the metric, and because it seems that it would be very hard to recover

standard unitary quantum mechanics at the nuclear and atomic level if the underlying

theory of quantum gravity were not unitary (cf. Section 4.2). He therefore advanced as an

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Ansatz the claim that the scattering of particles off a black hole – i.e., the absorption of

particles by a black hole and the subsequent emission of Hawking-like radiation – can be

described by a unitary S-matrix.

As a toy model of the behavior one might expect from such black hole scattering,

’t Hooft introduced a “brick wall model” of black hole scattering in his (1985). He begins

with a set of quantum matter fields, Ni(x), defined on a Schwarzschild black hole

spacetime. He then imposes both an infrared cutoff, by putting the system in a “box”of

radius L,

Ni(r) = 0 if r $ L,

and an ultraviolet cutoff requiring the fields to vanish within a distance h from the event

horizon at r = 2M,

Ni(r) = 0 if r # 2M + h.

’t Hooft then goes on to demonstrate that even though the quantum evolution of this

model is completely unitary, it will still generate Hawking-like radiation at late times.

Although this model does display the desired features of being actually unitary

while reproducing black hole thermodynamics at an effective coarse-grained level,

’t Hooft admits that it is not a plausible description of the actual evolution of black holes

for a number of reasons. First, scattering off of black holes should permit a number of

transitions that are not captured by the brick wall model. For example, it is generally

agreed that black hole formation and evaporation will violate baryon number

conservation, which is respected by ’t Hooft’s toy model. More significant for our

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purposes, however, is the fact that this model seems to be a version of the quantum

bleaching scenario. Because the field values can only be zero inside the “brick wall” at

r = 2M + h, it seems impossible to fall into a black hole, regardless of how small the local

curvature at the horizon may be.

While such a consequence may not seem overly significant in an obviously

preliminary toy model of this sort, ’t Hooft’s more refined proposal threatens to share this

problematic feature. This proposal begins with his S-matrix Ansatz: we assume that there

is an S-matrix that unitarily takes us from asymptotic in-states to asymptotic out-states. If

we begin in a pure state, therefore, the outgoing radiation will be in a pure state,

which only appears to behave like a mixed state as soon as we neglect to hand inall existing information about the ingoing particles. Averaging over the variouspossible modes for the ingoing particles will give us back the mixed state of theoutgoing objects. (’t Hooft 1996, p. 4653)

That is, the derivation of thermal Hawking radiation was imprecise in that it ignored the

interactions between the infalling matter and the outgoing radiation. One typically argues

that it is legitimate to neglect these interactions because one can demonstrate that the

changes to the Hawking state, DH, due to variations in the infalling matter are small.

However, ’t Hooft argues that if we assume that the final state is some unknown pure

state |out,, rather than the thermal Hawking state DH, then we will come to the conclusion

that interactions between the infalling and outgoing matter do play an important role in

determining the late-time state.

He therefore suggests that to construct the S-matrix we should begin with some

asymptotic in-state |in0, and an arbitrarily chosen out-state |out0,. This evolution will be

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assigned some transition amplitude +out0|in0, = A. We then consider a small adjustment to

the state |in0,, e.g., adding a single ingoing particle, and try to calculate the effect that this

change, *in, will have on the out-state. That is, we will try to calculate a *out so that we

can compute the S-matrix element +out0 + *out|in0 + *in,. The hope is that by calculating

the effects of these small changes on the outgoing radiation state we can eventually

generate the entire S-matrix.

To evaluate the effect of the extra ingoing particle on the outgoing particles, we

would, in principle, need to take into account all possible interactions between them.

However, ’t Hooft argues that the most significant interaction between infalling and

outgoing particles will be gravitational. In the frame appropriate to a late-time external

observer detecting Hawking-like radiation, the ingoing particle represented by *in will be

“boosted to such high energies that it produces a non negligible gravitational field”

(’t Hooft 1996, p. 4652). This gravitational field will generate a shift in the outgoing

particles, *y, indicated in Figure 5.1, which is copied from ’t Hooft (1996) p. 4652.

If the extra infalling particle has, in lightcone coordinates, a momentum of *pin, and a

transverse position x#N then the outgoing particles at transverse position x# will be shifted

inwards an amount

*xout = f(x# ! x#N)*pin.

Here f(x# - x#N) is a Green’s function satisfying

MT2 f(x# ! x#N) = !*2(x# - x#N),

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Figure 5.1: Shift of outgoing particle (’t Hooft).

where MT is the partial derivative with respect to the transverse coordinates, and *2 is a

two-dimensional delta function.

’t Hooft then goes on to introduce a momentum density operator P$ out(x#) that

generates the shift *xout. This allows us to represent the shift in the outgoing wave

function as:

He proposes that we can generate all in-states from |in0, by specifying the functions Pin(x#),

and, similarly, specifying Pout(x#) will enable us to describe all out-states. Our scattering

matrix elements will then be given by:

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where K is a normalization factor that will be fixed by requiring the S-matrix to be

unitary.

We shall not pursue the details of how ’t Hooft hopes to calculate these matrix

elements more specifically, nor the sizable challenges that this proposal faces both in

making realistic calculations of gravitational interactions and in incorporating other

forces. Instead we turn to the question of how he intends to reconcile the existence of his

unitary S-matrix with the premises that drive the information loss argument. We have

seen that he denies that we can trust the derivation of the late-time Hawking state, DH, and

that he expects the interactions between infalling and outgoing particles to safeguard the

purity of the external state. This might give us the impression that he is offering a version

of the bleaching scenario. However, he also explicitly rejects Hawking’s picture of an

evaporating black hole (see Figure 4.1), arguing that the existence of a true singularity in

these spacetimes would imply the loss of quantum coherence. The absence of a

singularity may then somehow allow the information to return from behind the horizon of

the black hole, and thus perhaps we can avoid the problematic bleaching solution.

’t Hooft suggests that whereas the in-states are naturally defined on a black hole

spacetime (Figure 5.2a), the out-states should be defined on a time-reversed white hole

spacetime depicted in Figure 5.2b. If we are to have a single metric that is suitable for

discussing both these sets of states, we will need to sew together the black hole and white

hole solutions in a singularity free manner. To do this we find a suitable time slice

J = Jmatch that we will identify in the two spacetimes, throwing out the regions of

spacetime later than Jmatch in the black hole case, and earlier than Jmatch in the white hole

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Figure 5.2: ’t Hooft’s spacetime for evaporating black hole.

case. The result is depicted in Figure 5.2c. To facilitate this identification we have

chosen our in- and out-states to be single shells of matter. This configuration of outgoing

Hawking radiation will be statistically unlikely, but it is one possible outcome of black

hole evaporation and it eases computation. The matter thus approaches the event horizon

and then “bounces” into an outgoing shell of matter.

Note that the dot-fill regions inside the horizons of Figures 5.2a and 5.2b are

inside the collapsing matter. Because this matter, like the spacetime itself, is spherically

symmetric, these interior regions are flat and can be joined without difficulty. Likewise,

the external regions of the two spacetimes can be joined together in a smooth manner.

The only problematic area of the spacetime depicted in Figure 5.2c is region S which will

be singular. However, ’t Hooft argues that this conical singularity is benign:

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It is of crucial importance to observe that the singularity is so mild that no loss ofquantum information is suffered by the evolving states of soft particles in such ametric. Indeed, if we replace the shells by more smoothly distributed matter thesingularity is smeared into a non-singular (but still highly curved) metric. Clearlythen, the S-matrix, in terms of the soft particles alone, will be unitary. (Stephens,’t Hooft, and Whiting 1994, p. 636)

“Soft particles” are, by definition, particles whose energies are small enough that their

individual gravitational effects can be neglected.

Our escape from the Hawking paradox therefore seems to rest in the rejection of

the premise that there is a true black hole singularity. Note, however, that there is still a

sense in which there is a black hole. Just as in the case of the remnant solutions discussed

in Section 4.4, we have here an example of irreversible gravitational collapse. The

S-matrix describing scattering off a black hole will allow the “information” concerning an

infalling astronaut to be returned to the external universe, but presumably in a highly

scrambled form. That is, we expect that the external universe will be left in some pure

state, which, given an ensemble of such states (all identically prepared), could in principle

be determined experimentally. This late-time pure state would then allow us to retrodict

the original state of the infalling astronaut, assuming we had already determined the S-

matrix appropriate for such a black hole. However, we should not expect an astronaut to

be spewed out of the black hole, except a vanishingly small percentage of the time. Thus

while ’t Hooft’s S-matrix does strictly deny the existence of any region that is not in the

causal past of I+, it does not deny the existence of black holes more broadly construed.

The fact that the early region of such a universe is isomorphic to a true black hole

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spacetime is a further reason for allowing ourselves to describe this evolution as the

formation and evaporation of a “black hole.”

5.2: Incompatible Field Measurements in the S-Matrix Ansatz

While ’t Hooft’s suggestion promises to offer an avenue of escape from the

information loss paradox, it is still not clear that it avoids the problems facing the

bleaching scenario. If the scattering of infalling particles, or the transfer of “information”

to outgoing Hawking-like radiation, occurs outside the apparent horizon (indicated by a

dashed line in Figure 5.2c), then it would appear that an infalling observer could not pass

through the horizon without being destroyed. However, if the scattering, or information

transfer, takes place inside the apparent horizon, then it does not seem that there would be

a sufficient number of particles to restore the purity of a late-time state, unless the rate of

Hawking-like radiation slowed considerably (recall the discussion of Section 4.4). In

other words, limiting the scattering events to the interior of the apparent horizon seems to

lead us to some sort of remnant scenario, which ’t Hooft specifically rejects.

He responds to this worry by claiming that measurements on the outgoing

Hawking radiation and measurements on the infalling matter performed inside the

apparent horizon are measurements of incompatible observables represented by

noncommuting Heisenberg picture operators – despite the fact that these observables are

spacelike related. This incompatibility is supposed to be grounded in the fact that the

commutators between operators, ÔH(t1), “describing any of the features of the outgoing

Hawking particles, such as their number operator, energies, correlations, etc.,” and the

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longitudinal component of the stress-energy tensor in light-cone coordinates, TÎ !!(hS),

grows exponentially with time (Stephans, ’t Hooft, and Whiting 1994, p. 625). It is worth

quoting at length the consequences that are claimed to follow from this fact.

In itself, this uncertainty relation would not have been a disaster if the particlescausing the large TÎ !! had been completely transparent. But they are not, becausethey must be associated with a gravitational field which, because of the infiniteenergy shifts involved, has the ability to destroy everything attempting to cross thehorizon, even if that crossing is to take place at different angular coordinates. Thus, we conclude that one cannot describe Hawking particles while at the sametime one describes observables, i.e. expectation values of local operators, beyondthe horizon. The corresponding operators have commutators which are far toolarge. One must choose the basis in which one wishes to work: either describeparticles beyond the horizon or the particles in the Hawking radiation, but do notattempt to describe both. Physically this means that one cannot have “superobservers”, observers that register both Hawking radiation and matter across thehorizon. The corresponding operators have explosive commutators. (Stephans, ’t Hooft, and Whiting 1994, p. 626)

The claim here is that the incompatibility of observables localized inside the black hole

and observables localized outside the black hole at late times, is due to the large

commutators between the late-time external observables and the stress-energy operator at

the horizon. ’t Hooft seems to be suggesting that a late-time measurement will,

apparently quite literally, destroy all infalling observers and measuring apparatus. This

claim is reiterated in his later overview of the S-matrix approach (I reproduce ’t Hooft’s

figure in Figure 5.3):

An observer A passes through an horizon, while also an observer B detectsHawking radiation. If this were flat space-time, these two observers wereconsidered to be spacelike separated, and therefore their measurement operatorscommute. Hilbert space can be factored into a space of states whose propertiescan be detected by A, and another space of states whose properties can be detectedby B, and possible further factors that can be seen neither by A nor by B. If,however, this space were considered to be the horizon of a black hole, the statesseen by A would be related to the states seen by B through an S-matrix, and hence

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no longer independent. For the black hole physicist, there is no contradiction. Anymeasurement made by B, implies the introduction of states obtained from the Hartle-Hawking vacuum by acting on it with operators that create or remove particles seen by B,which for A would be outrageously energetic. These particles would cause gravitationalshifts that seriously affect the ingoing objects, including the fragile detectors used by A. Thus these observations cannot be independent. What is new here, even for any possibleflat space-time observer, is that trans-Planckian particles are involved (with this term wemean particles whose energies are far beyond the Planck value). (’t Hooft 1996, p. 4684)

Taken at face value, the claim that a late-time measurement “implies the

introduction of states” that contain particles that are “outrageously energetic” according to

the infalling observer appears to be false. ’t Hooft’s argument seems to be based on an

interpretation of measurement collapse which, as I shall argue shortly, is illegitimate.

Further, the claim that the large commutators between TÎ !! and some late-time Hawking

observable ÔH(B) implies that ÔH(B) will fail to commute with observables inside the

horizon also appears to be mistaken. The failure of independence involved in this case

should involve mere correlations (in either the classical or quantum sense) and not a

failure of commutivity.

Figure 5.3:Early- and late-time observers.

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The picture that he seems to have in mind rests on the description of Hawking

radiation we considered in Section 3.4. We began in a global state that a radially infalling

observer would find indistinguishable from a Minkowski vacuum state. This implied that

such an observer would detect no particles whatsoever, even when passing through the

black hole horizon. A late-time “Rindler” observer, however, would detect thermally

distributed particles coming from the black hole. Let us suppose that this observer

detects one such particle at time t with energy T. If we evolve this particle back in time

to the point on the horizon where it meets the path of the infalling observer, we find that

it has extremely high energy on the order of Tet. In leaving the gravitational well of the

black hole, the particle’s energy was red-shifted down to the relatively small value T

recorded by our late-time observer. It therefore appears that this observer can conclude

that this highly energetic particle would have destroyed any observer trying to enter the

black hole. Although the mixed Hawking state, DH, is compatible with – indeed, it is

derivable from – a vacuum state experienced by an infalling observer, it would seem that

any actual measurement outcome is incompatible with such a vacuum. We should note

that assigning values to a complete set of commuting observables at a late time would

imply that the late-time state is in a pure state. This pure state, according to ’t Hooft’s

argument, is incompatible with the infalling observer passing safely through the horizon.

However, this picture appeals to an untenable account of measurement collapse.

The easiest way to see this is to recast ’t Hooft’s argument in the Schrödinger picture,

18 We expect this to be true as long as there is a well-defined Schrödinger picture. In thecontext of QFT in curved spacetimes this will not always be the case, because if therepresentations of algebras of observables at different times are not unitarily equivalent,then a Schrödinger picture of the evolution will not generally exist. However, ’t Hooftcannot appeal to these sorts of worries because he explicitly assumes that evolution fromearly to late times is unitary.

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which should be equivalent to the Heisenberg picture for all questions of principle.18 The

Schrödinger picture is also the most natural picture to use in this situation because, to the

extent that such a description is unproblematic, collapse will be a change in the state we

ascribe to the system; it thus make sense to use the picture in which time dependence is

restricted to the state. One primary problem with an interpretation of quantum mechanics

that rests on measurement collapse (or von Neumann’s “type one” evolution) lies in

specifying what sort of interactions are to count as a measurements. We can bypass this

problem here, however, and grant for the sake of argument that there are some procedures

that count as measurements and which induce a discontinuous change in the global state

of the field.

In the Heisenberg picture the causal relations underpinning time evolution are

captured by the nonvanishing commutators between operators representing timelike and

lightlike related observables. In the Schrödinger picture of particle mechanics and field

theory, all operators representing spatially separated observables will commute, and the

time dependence will be given by the evolution of the state. Thus the claim that TÎ !!(hS)

and Ô(xlate) have a large commutator will go over to the Schrödinger picture claim that a

state that is an eigenstate of TÎ !! at a time th, corresponding to hS, will evolve into a late-

time state that has a large dispersion for Ô. This implies that if we perform a TÎ !!

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measurement at th the state will change in such a way that we will be able predict very

little about the late-time Ô measurement.

Whatever criterion we adopt to decide when the measurement collapse occurs, it

is clear that at time th either a TÎ !! measurement has occurred or it has not. Thus the state

is either given by unitary Schrödinger evolution or it has discontinuously collapsed onto

an eigenstate of TÎ !!. However, neither of these possibilities poses any problem for the

health of an infalling observer, and there seems to be no reasonable mechanism by which

a measurement by a late-time Hawking observer (or “Rindler” observer, to use ’t Hooft’s

term) could possibly affect the happenings near the horizon. The non-vanishing

commutator between the Heisenberg picture operators indicates that the late-time

observer would, in principle, be able to tell whether or not the infalling observer

performed a measurement of the stress-energy of the field. It does not indicate that the

actions of the late-time observer can change the past and destroy the infalling observer.

We therefore have not been provided with a reason to believe that measurements

located inside and outside the black hole will be incompatible, and we are left without a

plausible response to the charge that the S-matrix Ansatz commits one to a version of the

bleaching scenario. In the past several years, however, ’t Hooft has offered a somewhat

different suggestion of how we should view information loss in black holes and the

unitary evolution of quantum field theory. In his 1999 article “Quantum Gravity as a

Dissipative Deterministic System,” he argues that black hole formation and evaporation

will indeed involve loss of information at a fundamental level. Further, and perhaps

surprisingly, he suggests that the evolution at this fundamental level will be deterministic.

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However, there will be a quantum level of description at which the evolution will be

unitary (i.e., information preserving) and will include the stochastic transitions of

standard quantum theory.

The essence of the suggestion is that one can suppose that there is a deterministic

evolution that is not time reversible because, for example, two distinct early-time states

are allowed to evolve into a single late-time state. If one then coarse-grains the

description by considering the equivalence class of states that evolve into a single late-

time state, then it is possible to have a time-reversible (information-preserving) evolution.

Clearly, this coarse-graining may also introduce a stochastic element into our description

because we have forgone an account of some of the details of the system.

While he does not claim to be able to offer many details of a possible underlying

deterministic theory of quantum gravity, ’t Hooft does suggest some features that we

might expect from such a theory. He pictures a theory formulated in a Hilbert space in

which there will be a “primordial basis” of the “beables” of the system. The term

“beable” is borrowed by ’t Hooft from John S. Bell, who introduced the term to pick out

the properties that have robust ontological status, by contrast to the “observables” of the

system, which, although straightforwardly specifiable as being represented by Hermitian

operators, rest “physically [on] a rather wooly concept” (Bell 1987, p. 54). ’t Hooft, on

the other hand, contrasts beables with “changeables,” by which he means operators that

“may replace a state by a different state, in a different equivalence class,” where these

equivalence classes are the those that allow us to move from our deterministic dissipative

description to our time-reversible stochastic one.

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Beables on this account will be Hermitian operators that, in the Heisenberg

picture, all commute with each other at all times – while changeables will in general not

so commute. Presumably this characterization is motivated by the view that the true facts

about the deterministic system will have to be definite, which seems to imply that the

actual state of a system should be dispersion free for any operator that represents a beable.

Because the theory is deterministic, in the Heisenberg picture the state should fix the

precise values of the beables for all time – thus any possible state will have to be an

eigenstate of all beables. This in turn seems to imply that the beables will all have to

mutually commute.

However, this characterization of beables is somewhat unexpected for at least two

reasons. The first is that the paradigmatic example of a beable (and the example that Bell

surely had in mind) is that of position in Bohm’s nonlocal hidden-variable theory. The

Heisenberg-picture operators that are the quantum representation of this beable will not

generally commute. The reason for this is that the definiteness of position in Bohm’s

theory does not require the system to be in a position eigenstate: the actual property is not

represented by a state in a Hilbert space, but by a “hidden variable” – a point in

configuration space. The second reason that ’t Hooft’s characterization is surprising is

that properties represented by Heisenberg-picture operators that commute between

themselves at all times generally represent constants of the motion. (Indeed, most of the

examples of beables that ’t Hooft considers in his toy models are indeed constant under

time evolution.) While there are some examples of operators that evolve in such a way

that when considered at any two times they will commute with themselves, it is not clear

19 Such examples are discussed in the literature on Quantum Non-Demolition (QND)measurements. See, for example, Caves et al. (1980). ’t Hooft also introduces such anoperator in his model of non-interacting massless fermions to form a completecommuting set of operators with two other operators, both of which are constants ofmotion (’t Hooft 2001).

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that such examples are physically significant.19 Thus it is not clear that this

characterization of beables is likely to lead to a useful account of fundamental dynamics

of our system.

An obvious objection facing this proposal is the fact that the experimental

violation of the Bell inequality implies that any hidden-variable model (including a

deterministic model of the sort ’t Hooft is advocating) will have to be nonlocal. It is clear

that ’t Hooft is aware of this objection, but his response to it is less transparent. In his

1999 essay “Quantum Gravity as a Dissipative Deterministic System,” he argues that his

proposed deterministic theory will not conflict with Bell’s theorem because it only

applies at the Planckian scale, but only standard quantum theory can be applied to

describe the phenomena involving the electrons or photons that we use to test the Bell

inequality:

Bell’s inequalities follow if one assumes deterministic equations of motion to beresponsible for the behaviour of quantum mechanical particles at large scales. . . .In our theory, however, the wavefunction has exactly the meaning andinterpretation as in usual quantum mechanics; it describes the probability thatsomething will or will not happen, given all other information of the systemavailable to us. ‘Reality,’ as we perceive it, does not refer to the question ofwhether an electron went through one slit or another. It is our belief that the truedegrees of freedom are not describing electrons or any other particles at all, butmicroscopic variables at scales comparable to the Planck scale. Their fluctuationsare chaotic, and no deterministic equation exists at all that describes the effects ofthese fluctuations at large scales. Thus the behaviour of things we call electronsand photons is essentially entirely unpredictable. It so happens, however, thatsome regularities occur within all these stochastic oscillations, and the only way to

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describe these regularities is by making use of Hilbert space techniques. (’t Hooft1999, p. 3272, emphasis original)

It is difficult to decipher this suggestion. On the one hand, it seems that ’t Hooft is

merely claiming that the quantum description is a coarse-grained account that is

stochastic because it fails to include all the details of the actual deterministic system

underlying the phenomena. That is, QM – rather like thermodynamics – offers an

account of “the probability that something will or will not happen, given all other

information available to us” (emphasis added). On the other hand, it seems that ’t Hooft

is claiming that it will not merely be beyond our abilities to use the fundamental

deterministic theory to describe large-scale phenomena, but the theory will not even be

valid at these scales. This seems to be a point of principle about the domain of validity of

the theory: “no deterministic equation exists at all that describes the effects . . . and the

only way to describe these regularities is by making use of Hilbert space techniques,” by

which ’t Hooft presumably means the standard quantum theoretic account (some

emphasis added in quotation).

The former reading, of course, provides no refuge from Bell’s theorem – unless it

somehow opens the door for the deterministic theory to be nonlocal, a possibility we shall

consider in a moment – for Bell’s theorem specifically addresses the possibility of

hidden variables that might supplement the quantum theory. It is less obvious that we can

reject the second reading, but this is largely because it is less obvious what this

suggestion might amount to. At some points ’t Hooft seems to believe that Bell’s

20As Jim Cushing was fond of pointing out in conversation, “Bell’s theorem has nothingto do with quantum mechanics.” Of course, the theorem is interesting largely because itsinequality is violated by quantum theory, but the fact that experimental results violate thisinequality is entirely independent of the representation of those phenomena by quantummechanics.

21We cannot take the time here to consider what sort of violation of locality (as Bell usesthe word) is involved here. The interested reader is referred to Cushing and McMullin(1989) and references therein for a discussion of different characterizations of locality or

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theorem can be avoided because the deterministic theory will not make direct reference to

quantum entities.

When we measure the spin of a photon . . . our measuring instrument is as much achaotic object as the phenomena measured, and only at macroscopic scales can wedetect statistical regularities that can in no other way be linked to microscopicbehaviour other than by assuming Schrödinger’s equation. The idea that theremight exist a deterministic law of physics underlying all of this essentiallyamounts to nothing more than the suggestion that there exists a ‘primordial basis,’a preferred basis of states in Hilbert space with the property that any operator thathappens to be diagonal in this basis, will continue to be diagonal during theevolution of the system. None of the operators describing present-day atomic andsubatomic physics will be completely diagonal in this basis. This enables us toaccept both quantum mechanics with its usual interpretation and to assume thatthere is a deterministic physical theory lying underneath it. Apparently, we areforced to deny the existence of electrons, and other microscopic objects, even ifthey appear to be obvious explanations of observed phenomena. (’t Hooft, 1999,p. 3272, emphasis original)

However, merely denying the operators of current quantum theory a fundamental status,

or even denying the existence of electrons and so on, does not lessen the force of Bell’s

argument against local deterministic theories. Bell’s theorem does not depend on a

description in terms of electrons, nor does it have any commitment to the operators of

standard quantum theory.20 If a deterministic description is accurate (and all the usual

auxiliary hypotheses are satisfied – see, e.g., Wessels, 1989) then we are committed to

some sort of violation of locality.21 The fact that the lessons of Bell’s results are logically

separability that might be taken to be refuted by the Bell results.

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independent of any quantum theoretic description also undermines the relevance of a

rejoinder claiming that although the “ontological states may still evolve according to a

completely local law . . . the equivalence classes are not locally well-defined” (’t Hooft,

2002, p. 314). The definability of local quantum observables or states is irrelevant; we

only need the claim that the outcomes of our Bell-type experiments are accurate and well

localized for us to rule out a local deterministic account of them.

If ’t Hooft did not explicitly claim that his deterministic theory is also a

local theory, we might suspect that he is advocating the possibility of a breakdown of

local descriptions at the Planck scale along with the claim that a local quantum

description can only apply at a coarse-grained stochastic level. Likewise, his claim that

both quantum theory (“with its usual interpretation”) and the deterministic theory are

valid might lead us to wonder whether he is advocating some very strong theoretical

pluralism, which claims that there is no reductive relationship between the quantum

description and the deterministic Planckian one. But this reading seems to be ruled out

both by ’t Hooft’s presentation of the deterministic theory “lying beneath” the quantum

one, and by his suggestion that the quantum description should be derivable from the

Planckian one by grouping fundamental states into equivalence classes.

In a paper dedicated to Bell, ’t Hooft takes a different line and suggests that

perhaps we should not believe that we have true experimental confirmation of violations

of the Bell inequalities – for perhaps the distant measurement settings are not actually

independent. Such suggestions are typically discounted because each experimenter can

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presumably freely decide how to set the apparatus (and thus we can eliminate any

correlations between the distant detector settings). But ’t Hooft suggests that this

impression of freedom might be illusory:

The experimenter’s decisions also follow from Standard Model interactions in hisbrain. Predeterminism is here defined by the assumption that the experimenter’s‘free will’ in deciding what to measure (such as his choice to measure the x- or they-component of an electron’s spin), is in fact limited by deterministic laws, hencenot free at all, so that it must be included in a deterministic interpretation ofQuantum Mechanics. (2002, p. 316)

While this route certainly would allow one to reconcile a local deterministic theory with

Bell’s theorem, at the moment it seems far fetched and poorly motivated. Leaving aside

the issue of free will, the setting of the distant apparatuses can be set by an extremely

complex, but still perfectly-well understood, mechanism. To use Bell’s example, we

might decide the setting of the measuring device based on whether the millionth decimal

place of some variable is even or odd (Bell 1987, p. 103). It seems unreasonable to

believe that there could be a plausible subquantum theory that might conspire to

manipulate all the necessary features of the system to reproduce quantum statistics.

We should note that ’t Hooft does claim that the quantum mechanical description

will eventually break down because the true fundamental processes are deterministic.

However, he places this proposed limit far beyond the reach of our current experiments:

the limit lies in a bound to the ability of quantum computers to exceed the ability of

classical computers (’t Hooft 1999, p. 3273). It is far from clear, however, why there

should be a bound to the quantum description here and not in tests of the Bell

inequalities, the two slit experiment, and so on. One might guess that the justification lies

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in a commitment to information being a fundamental concept in quantum gravity, as

opposed to locality, for example; but this remains mere speculation, for ’t Hooft has yet to

articulate a clear account of why quantum computation is the (only?) quantum-like

behavior that cannot be successfully mimicked by the fundamental local deterministic

theory.

We therefore do not yet seem to have compelling reasons for relaxing our

suspicion of a local deterministic theory of the sort ’t Hooft proposes. More relevant for

our purposes here, however, is the fact that it is not clear how such a deterministic

dissipative description would support ’t Hooft’s earlier suggestions about the

noncommutativity of operators representing observables inside and outside the black

hole. Let us therefore turn to an account that was inspired by ’t Hooft’s earlier arguments

and that brought BHC to the attention of most physicists working on quantum gravity.

5.3: Nice Slices and Three Postulates of Black Hole Complementarity

The 1993 paper “The Stretched Horizon and Black Hole Complementarity” by

STU (Susskind, Thorlacius, and Uglum) attempts to respond to Hawking's information

loss argument by offering a foundation for a “phenomenological description of black

holes” that is explicitly unitary (p. 3744). To say that the description is intended to be

phenomenological is to say that it is intended to be an effective description of the black

hole’s evolution; it does not pretend to offer a complete account of the sort that we might

expect from a full theory of quantum gravity. Nevertheless, they hope that their model

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will provide insights into such a full theory, by offering an alternative to the standard

effective description that drives the information loss paradox.

The heart of their proposal is the claim that a black hole should not, from the

perspective of a low energy theory appropriate to an external observer, be described in

terms of local fields defined on the black hole region. Instead, such a theory should

describe a quantum mechanical membrane just outside the event horizon. This allows the

evolution describing the formation and evaporation of the black hole to be completely

unitary, for in this description the “information” never passes into the inaccessible region

behind the event horizon. This is precisely the “bleaching” proposal discussed in Section

4.2, except for the proviso that this description is merely an effective description. We

shall see below how the complementarian (i.e., the advocate of black hole

complementarity) appeals to this fact in response to the objections raised against the

bleaching scenario.

STU lay out the essential elements of their proposed theory in three postulates, the

first of which is in essence an endorsement of ’t Hooft’s S-matrix Ansatz.

Postulate 1. The process of formation and evaporation of a black hole, as viewed by adistant observer, can be described entirely within the context of standard quantumtheory. In particular, there exists a unitary S-matrix which describes the evolutionfrom infalling matter to outgoing Hawking-like radiation. (p. 3743)

This asserts that the evolution of a black hole will be effectively unitary; in particular, if

we begin in a pure state we will end in a pure state. It also specifies that the information

will be carried away by the Hawking-like radiation (recall from Section 4.3 that true

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Hawking radiation is a thermal mixed state, and it therefore cannot unitarily evolve from

a pure state), and is not retained in a black hole remnant.

The second postulate refers to the stretched horizon of the “membrane paradigm”

of black holes discussed in Section 3.1. The stretched horizon considered here is a

surface just outside the event horizon that has an area one Planck area larger than the

event horizon. This surface can be treated by an external observer as a physical

membrane carrying all the properties of the black hole.

Postulate 2. Outside the stretched horizon of a massive black hole, physics can bedescribed to good approximation by a set of semiclassical field equations. (ibid.)

The approximate validity of semiclassical gravity is, of course, a key component of both

the argument for nonunitary evolution, and for the remnant proposal's claim that

information can only escape after the black hole has shrunk to Planck size. The crucial

difference in STU's postulate is the restriction of the validity of semiclassical field theory

to the region outside the black hole. The question of whether this restriction can be

justified is central to the debate between the complementarians and the advocates of

remnants or information loss.

The first postulate describes the global evolution of the quantum system, the

second tells us how to describe the region outside the black hole, and the third gives us a

prescription for effectively describing the black hole itself.

Postulate 3. To a distant observer, a black hole appears to be a quantum system withdiscrete energy levels. The dimension of the subspace of states describing a blackhole of mass M is the exponential of the Bekenstein entropy S(M). (p. 3744)

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Here we see the assumption of the validity of black hole thermodynamics coming into

play. The number of degrees of freedom that are postulated as being properly attributable

to the black hole is quite a large number, eS(M), where S(M) is given in units of Planck area

(cf. Section 3.6), but it is still considerably smaller that the infinite number of degrees of

freedom that a field theory would ascribe to the same region. However, we should also

recall that an effective theory with a high energy cutoff in place will also attribute a finite

number of degrees of freedom to any finite region. An important difference between such

a theory and the proposal being put forward by STU, however, is that Postulate 3 implies

that the effective number of degrees of freedom of a black hole will grow as the area of

the black hole, and not as the volume as a field theory would predict. This observation is

the foundation of the Holographic Principle, which we shall discuss in Section 6.1.

The proposal, then, is that an external observer can treat a large black hole as a

physical membrane just above the event horizon of the black hole, where this membrane

can be considered a quantum system with a very large, but finite, number of degrees of

freedom. This stretched horizon will interact with the quantum field that, according to

Postulate 2, can be described by a semiclassical QFT outside the stretched horizon. The

state of our total system, according to this effective low energy description appropriate to

an external observer, can be described as lying in a Hilbert space ,sh q ,ext, where ,sh

corresponds to the eS(M) degrees of freedom associated with the stretched horizon, and ,ext

describes the external effective quantum field theory. Correlations will be established

between the two subspaces as the black hole is formed and evolves, but as it gives off

Hawking-like radiation it loses mass and the size of ,sh shrinks accordingly. As the

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black hole shrinks, the correlations will be passed off to the outgoing radiation. Because

the membrane is outside of the true event horizon, this picture of the restoration of

information to the external universe does not involve any explicit superluminal

information transfer. Once the black hole has completely evaporated away, all of the

correlations are contained in states of the external quantum fields, and the entropy of the

global state is unchanged.

STU do not offer a specific, realistic model of semiclassical gravity that would be

appropriate for an external observer to use. However, they do offer a highly idealized

two-dimensional toy model that they hope will capture the key features of a full

description of a black hole’s evolution. We relegate the details of this model to Appendix

B, and merely report some of its key features here. A shortcoming of the semiclassical

theory of quantum gravity given by Equation (4.1) is that we are unable to solve these

equations. STU’s model, on the other hand, is carefully (if somewhat unnaturally)

adjusted to yield solutions.

To generate a solution we must impose boundary conditions at the singularity at

the center of the black hole. This allows us to follow the evolution of a black hole giving

off Hawking radiation and shrinking. However, this model faces problems including the

presence of a naked singularity at the end of evaporation and potential violations of

energy conservation. This offers one of the primary motivations for STU to adopt the

membrane paradigm of black holes: if the boundary conditions are applied at a stretched

horizon just outside the event horizon, rather than at the singularity, then the theory

arguably offers a physically reasonable description of the spacetime outside the stretched

22 Two further points along these lines will be developed in the next chapter. First, wehave a rough picture of the expected process from string theory, and this picture seems toimply strong destructive interactions at the horizon (from the external observer’s

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horizon. The semiclassical theory mentioned in their second postulate would be a more

realistic theory that shares the basic features of this two-dimensional model.

According to their third postulate, an external low-energy observer should then be

able to describe the stretched horizon as a quantum system with a number of degrees of

freedom given by the Bekenstein entropy. STU do not suggest any details of the

dynamics that would govern this membrane, but they are committed to the claim that the

global evolution of the system will be effectively describable – by the external observer –

as unitary evolution of a state in a Hilbert space given by ,shq,ext. Such an observer

could legitimately say that any correlations carried by matter falling into the black hole

are passed off to the stretched horizon, and are then returned in scrambled form to the

external field in the Hawking-like radiation. Thus the possibility of unitary evolution is

saved and Hawking’s paradox is avoided.

Notice that there are two expectations that lead the complementarians to believe

that their postulates commit them to the destruction of any observers that fall into the

black hole. The first is that they accept Hawking’s prediction of the of the outgoing

radiation to be a legitimate effective description of the situation. Therefore the observers

must be destroyed since observers are not even approximately thermal radiation. The

second is that they are committed to the claim that an infalling object will strongly

interact with eSbh degrees of freedom; such interactions are not likely to be compatible

with the survival of any recognizable object.22

perspective). Second, given that an infalling observer will attest to her own destruction atthe center of the black hole, consistency seems to require that she not be allowed toescape to live out her life with an external observer.

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The primary difficulty facing STU’s proposed effective theory is the obvious fact

that it seems to deny the possibility of any object passing through the event horizon

without being destroyed. If all of the information falling towards the black hole remains

at the stretched horizon where it can eventually escape out to infinity, then it seems that

an observer would be unable to pass into the interior of the black hole. However, as we

argued in Section 4.3, this “quantum bleaching” seems to be ruled out by the fact that the

event horizon, and hence the stretched horizon, is a globally defined feature of spacetime;

we expect that for a large black hole the horizon region will be locally indistinguishable

from any other region in spacetime.

STU respond to this objection by agreeing that “a freely falling observer

experiences nothing out of the ordinary when crossing the horizon” (p. 3744). The

question, of course, is how we are to justify this assumption in light of the argument for

information loss. STU’s strategy, which we shall investigate in more detail in the

following section, is to deny that one can offer a single combined description of the

interior and exterior of the black hole, “with a standard quantum theory, valid for both

observers” (ibid.). They continue,

Of course, it may be argued that a quantum field theoretic description of gravitydictates just such a description, whether we like it or not. If this is the case, such aquantum field theory is inconsistent with our postulates; therefore, one or theother is incorrect. (ibid.)

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However, it is not STU’s postulates that are in conflict with local QFT (unless we take

QFT to exclude the possibility of spatially extended objects, like our stretched horizon,

that have a finite number of degrees of freedom). The bleaching scenario is perfectly

compatible with QFT; it is only incompatible with the expected gravitational nature of the

event horizon. It is their denial of bleaching, while advancing their three postulates that

is incompatible with local QFT.

This incompatibility can be seen from the arguments we considered briefly in the

last chapter. Local QFT tells us that field operators are to be assigned to every point, or at

least to every bounded region, of spacetime. If the black hole is a region of spacetime, as

it must be according to the usual definition of black holes, then there must be field

operators associated with it, and the global state of the field will indeed by given by a

state D0 in the joint Hilbert space ,bhq,ext. The denial of bleaching implies that this state

will be correlated, and microcausality implies that these correlations cannot be passed off

to the outgoing Hawking radiation as STU envision. Postulate 3, which claims that the

distant observer can legitimately describe the black hole as a membrane with a finite

number of degrees of freedom that are accessible to the external world, is incompatible

with the claim that an observer can pass through the horizon into a region that is

describable by globally applicable QFT. Thus STU’s proposal entails a rejection of the

validity of QFT within the black hole.

We might find this rejection somewhat unsurprising in light of the fact that STU,

and most other black hole complementarians, are string theorists. As such, they expect

that the fundamental objects of quantum gravity will be nonlocal: a local description in

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terms of a field theory defined at points, or regions of arbitrarily small volumes, will only

be valid as a limiting case at low energies. Because QFT clearly is an extremely accurate

description of low energy situations, however, they cannot escape Hawking’s argument

by simply denying the universal validity of local QFT; they need to demonstrate that it is

reasonable to believe that the quantum field theoretic description breaks down in the case

of black hole evolution, and that this breakdown opens up a possibility for describing the

global evolution unitarily. The problem is that it appears on the face of it that local QFT

should offer an adequate description of all events occurring in the region around the event

horizon of a large black hole, and thus that it can still ground Hawking’s argument.

The argument for this claim plays a foundational role in justifying both the

information loss argument and the remnant proposal, although it is often not fully

articulated. Typically one points out, as we did above, that there is nothing locally

significant about the event horizon, and that the local spacetime curvature there can be as

small as we like for sufficiently large black holes. Therefore we do not expect any

quantum gravitational effects to come into play in this area. However, for this argument

to be convincing we need to be clearer about exactly what a “quantum gravitational

effect” would be, and about how we are to specify when we can, and cannot, legitimately

ignore these effects.

A more rigorous argument appeals to the fact, commonly raised in the context of

renormalization and effective field theories, that high energy degrees of freedom will

generally decouple when all energies involved are low – meaning that a description

involving only low energy degrees of freedom should adequately describe such a process.

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Because high energies map onto short distances, this seems to imply that the nonlocal

nature of strings – whose length is on the order of the Planck length – can only manifest

itself if a process includes Planck-scale energies. If it is possible to run the information

loss argument in a situation that avoids such high energies, then it would seem that the

effective validity of QFT would secure Hawking’s conclusion. Such a description of the

situation does seem to be possible, however, for we can construct a series of spacelike

time slices that avoid all regions of high curvature for as long as possible, and for which

both the infalling bodies and the outgoing radiation have low energies in the frame of the

time slice. Such foliations of spacetime are referred to as “nice slices.”

Because all of the local energies on nice slices are low, it seems that all processes

should be adequately described by a QFT obeying microcausality. Further, it is possible

to construct a family of nice slices that pass through the infalling matter and the outgoing

Hawking radiation for most of the evaporation of a large black hole. Therefore, proceeds

the argument, the information cannot escape when the black hole is large, but must wait

until it shrinks to Planckian size, and we can no longer define the state of the system as a

state of a nice-slice field. At this point we can expect Planckian physics to come into

play – but if these quantum gravitational effects are to rescue unitary evolution at this late

stage they will have to slow or halt the Hawking radiation and leave us with a remnant.

STU want to argue that this argument is flawed. The mere fact that one can

construct a low energy field theory on a nice slice does not imply that the nonlocal effects

of the underlying theory of quantum gravity cannot manifest themselves. The heart of

STU’s account of BHC is the denial of the claim that the validity of local QFT at low

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energies gives us sufficient grounds to agree with Hawking that the evolution of black

holes will be nonunitary.

5.4: Susskind, Thorlacius, and Uglum on Black Hole Complementarity

The challenge that STU face is that of justifying their claim that their effective

description in terms of a heated membrane at the stretched horizon is compatible with

their claim that an infalling observer would notice nothing out of the ordinary when

passing through this horizon. Because incompatibility of these two claims can be derived

from QFT, any adequate response will need to argue that the field theoretic description of

the scenario can somehow be dismissed. The core of STU’s attempt to dismiss the

effective QFT description lies in questioning the physical legitimacy of the field theoretic

joint state used to drive Hawking’s paradox.

The assumption of a state |Q(EP), which simultaneously describes both theinterior and the exterior of a black hole seems suspiciously unphysical. Such astate can describe correlations which have no operational meaning, since anobserver who passes behind the event horizon can never communicate the resultof any experiment performed inside the black hole to an observer outside theblack hole. The above description of the state lying in the tensor product space,bhq,[ext] can only be made use of by a “superobserver” outside our Universe. Aslong as we do not postulate such observers, we see no logical contradiction inassuming that a distant observer sees all infalling information returned inHawking-like radiation, and that the infalling observer experiences nothingunusual before or during horizon crossing. (p. 3744)

|Q(EP), is simply the state that, in section 4.1, we called D0; it is the state of the field on

the last foliation to include the interior and exterior of the black hole and is defined in the

joint Hilbert space ,bhq,ext. The central conceptual question that we face is how STU

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hope to motivate their dismissal of the description offered by this state, and whether this

dismissal has any interesting ties to Bohr’s account of complementarity.

We can disentangle three strands of justification that run through their arguments

for this claim. The first is an operationalist position, apparent in the above quotation, that

seems both problematic and foreign to Bohr’s account of complementarity. The second is

a tu quoque of sorts, arguing that the assumed validity of a local QFT description in the

context of black holes is an assumption standing in need of justification as much as their

own assumption that the suggested effective descriptions of the two observers is

compatible. The third line of argument tries to formulate a more substantial account of

the incompatibility of the measurements in question. Our discussion of the most

promising avenue along this line, which makes an appeal to string theory, will be deferred

to the next chapter. Here we shall focus on STU’s arguments for the impossibility of

sharing the results of the two measurements, and ask whether this can provide a

legitimate grounding for ruling out the possibility of a joint measurement of these

observables.

Let us begin with the operationalist move that dismisses the contradictory

description because it is unverifiable. Advocates of black hole complementarity typically

support their claim that the local QFT description is inapplicable by arguing that it would

be impossible, in principle, for any physical beings to collect the data to support the

predictions of such a field theory in the context of black holes. This is the substance of

STU’s claim that the correlations described by a state in the field theoretic joint Hilbert

space ,extq,bh “have no operational meaning.” They support this claim by pointing out

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that any measurement of a field value inside the black hole cannot be communicated to an

experimenter who remains outside the black hole to perform an external measurement.

Suppose, for example, Alice falls into a black hole with a collection of electrons each of

which is pairwise entangled in a singlet state with another collection that Bob keeps

outside the black hole. Standard quantum theory then tells us that the interior and the

exterior of the black hole are in an entangled state, which then allows us to run our

information loss argument. But STU ask whether it would be possible to verify the

predicted correlations between Bob measuring the spin of his electrons and Alice

measuring the spin of hers. Alice can apparently never communicate her findings to Bob,

because any messages she might send can never escape the black hole but are doomed,

along with Alice herself, to destruction at the central singularity. Therefore the

verification of the correlations could only be carried out by a “‘superobserver’ outside our

Universe.”

This operationalist move is problematic for a number of reasons. First, it is

unclear precisely what needs to be verified, according to STU. At one point they question

the possibility of testing for the quantum correlations encoded in the state D0; at another

they focus on whether the infalling observer’s report of the absence of a heated membrane

at the stretched horizon can reach an external observer; and at yet another point they

question whether the apparent violation of the prohibition on quantum xeroxing can be

violated. While there are obvious connections between these issues, their relevance

cannot be assessed until we have a clearer account of what argumentative force the

absence of “operational meaning” is supposed to carry.

23 See Quine (1951). As an example of a related critique of operationalism, see Gillies(1972).

24 Belot, Earman, and Ruetsche also make this point when they engage in some“considered grumbling” over STU’s appeal to operationalism (BER, p. 214). While Ishare BER’s view that this appeal is illegitimate, there is considerably more to be said forSTU’s arguments, as we shall see in the rest of this section. A weakness of BER’sarguments is that they do not seem to recognize that the complementarian is trying tooffer a merely effective description of the situation, and that the real force of thearguments, here as elsewhere in the debate over Hawking’s paradox, rests on claims overwhen and where our effective descriptions will break down.

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This leads us to the second problem with appealing to verificationism or

operationalism in this context: the view does not seem to be supported well

philosophically. The philosophical literature of the past sixty years contains a number of

well-known critiques of this account of meaning; for example, Quine famously argues

that meaning can only be given to theoretical terms when those terms are considered as

part of an entire web of theories and beliefs.23 A related worry is that of specifying what

is to count as admissible verification and when it is legitimate to invoke this

verificationist principle. It is noteworthy, for example, that STU apparently believe that

in the context of classical GR it is legitimate for an external observer to offer a theoretical

description of the interior of a black hole, even though such an observer could never

receive any reports from investigators who enter the hole.24 Should we say that the joint

state describing the outcomes of the internal and external observers is “unphysical” in this

case as well, since there would be no way for a physical being to verify the outcomes

described by the state? Indeed, if we did adopt this verificationist position, it would be

difficult to motivate the project of looking for a consistent interpretation of the interior of

25 In the quantum gravitational context we might also consider the case of two observersin a spacetime that is expanding sufficiently rapidly that the observers’ forward lightcones do not overlap. The presence of such observers (and the joint outcomes of theirmeasurements) could never be verified by any third physical being. It is unclear whatblack hole complementarians would want to say about this case. The position of space-time complementarity, discussed in the following section, might imply that theobservables associated with the apparatuses of such observers would indeed becomplementarity. Whether this could be supported, or reconciled, with a string-theoreticor M-theoretic description of the situation (see Section 6.1) is an interesting question thatgoes beyond the scope of this dissertation.

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black holes at all, to say nothing of the project of developing a description of Planck-scale

physics.25

A third problem with STU’s appeal to verificationism is the fact that it is not

immediately obvious that it would be impossible for a physical observer to have access to

the outcomes of an interior and exterior measurement. Shortly after they published their

1993 article, a number of objections were raised against this claim, objections that

Susskind and Thorlacius tried to answer in their 1994 article, “Gedanken Experiments

Involving Black Holes.” Their paper investigates “contradictions” that arise from the

claim that information falling into the black hole is stored on the stretched horizon, while

simultaneously asserting that an infalling observer experiences nothing unusual at the

horizon. The contradictions they have in mind here are those centered on the

impossibility of quantum xeroxing, or the “duplication of quantum information” (p. 970).

Their response echos the position of the STU paper: “So long as any single observer can

never learn the results of experiments performed on both copies of the quantum state we

are not led to logical contradiction” (Susskind and Thorlacius 1994, p. 970).

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There is an ambiguity in Susskind and Thorlacius’ presentation regarding whether

the scenarios described in these Gedankenexperimente are claimed to be impossible in

principle, or whether we simply cannot reasonably judge the results of such a scenario

without appealing to some theory of Planck-scale physics.

[O]bvious logical contradictions only arise when one attempts to correlate theresults of experiments performed on both sides of the event horizon. Theprinciple of black hole complementarity . . . states that such contradictions neveroccur because the black hole interior is not in the causal past of any observer whocan measure the information content of the Hawking radiation. (Susskind andThorlacius, 1994, p. 966)

If they truly hope to answer the charge that their account is contradictory, then it seems

that even if we allow them an appeal to verificationism they will have to establish the

claim that such communication is impossible in principle.

On the other hand, STU’s goal of offering a merely effective external description

of an evaporating black hole might make room for the claim that if one needs to appeal to

Planckian energies then one has left the realm of validity of our low energy theory. This

line of argument, which we shall investigate in more detail below, also fits well with the

stated goal of Susskind and Thorlacius’paper:

Our aim is limited to challenging the commonly held view that, as there is nostrong curvature or other coordinate invariant manifestation of the event horizon,an information paradox can be posed without detailed knowledge of theunderlying short-distance physics. (Susskind and Thorlacius 1994, p. 966)

This is the basis of the second line of argument, the tu quoque mentioned earlier. If the

advocates of remnants or information loss are forced to appeal to the nature of Planck-

scale physics in their arguments, then it seems that these arguments will only be

persuasive to one who shares their assumptions about the nature of a full theory of

26 The claim that the three postulates of BHC are consistent with an infalling observerexperiencing nothing out of the ordinary until she reaches regions of high curvature at thecenter of a black hole was elevated to the status of a principle, the “principle of black holecomplementarity,” in Susskind and Thorlacius (1994, p. 966).

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quantum gravity. If someone has different expectations from QG, then the argument for

information loss or remnants will not be compelling.

This line of argument leads us to expect a clear account of how Hawking’s

argument makes an unwarranted appeal to the nature of Planckian physics, and (perhaps

equivalently) an account of where the nice-slice argument goes wrong. What we find,

however, is that Susskind and Thorlacius seem to be merely admitting that it would, in

principle, be possible to verify the correlations that were ruled out by the STU paper, but

then arguing that such verifications inevitably require the experimenters to utilize

Planckian energies. They therefore conclude that “attempts to evade black hole

complementarity involve unjustified extrapolation far beyond the Planckian scale”

(Susskind and Thorlacius, p. 972). This is the basis for their above claim that one cannot

pose the information loss paradox without “detailed knowledge” of Planck-scale physics.

The problem with this argument is that it seems to appeal once again to a

questionable verificationism – all the more questionable in this case because verification

is not impossible, but is merely beyond the domain of applicability of our current physical

theories. It thus seems that the strongest conclusion we are justified in asserting is that

the verification in question might not be possible: our current theories cannot guarantee a

verification of the falsity of the “principle” of black hole complementarity.26 This is a

weak conclusion, and one that falls significantly short of Susskind and Thorlacius’ stated

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aim of demonstrating that Hawking’s argument cannot be posed without knowledge of a

full theory of quantum gravity. Before reaching judgment on the legitimacy of Susskind

and Thorlacius’ argument, however, we should consider the attempts to “evade black hole

complementarity” that are foiled by a need for Planck-scale energies.

Susskind and Thorlacius, like STU, worry about three types of observations that

might falsify their position. The first is the possibility of measuring quantum correlations

inside and outside the black hole. If such measurements are possible then we seem to be

forced to admit that the global correlated state is “physical,” and thus Hawking’s

argument seems to go through. The second worry is that the external observer might

learn that the infalling observer experiences no heated membrane at the stretched horizon.

The applicability of the membrane paradigm to black holes is an unproblematic

consequence of GR, and thus would seem to require no questionable verificationist

maneuvers, but the external observer’s description of an infalling observer interacting

with (burning up on) a Planck-temperature membrane and being reemitted in Hawking-

like radiation is somewhat harder to reconcile with the infalling observer’s perspective.

This is related to the third worry: According to the external observer the late-time

external state of the field will be pure (assuming that it began in a pure state), because all

correlations that fell into the black hole were passed off to the heated membrane and then

reemitted in Hawking-like radiation. The infalling observer, however, will claim that the

correlations could not have been passed off to any other system at the stretched horizon,

for the matter she is carrying did not interact with anything in this region of spacetime.

The claim that the correlations, or information, can be found both in the outgoing

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Hawking radiation and in the matter the infalling observer carries, then seems to violate

the prohibition on quantum xeroxing. If it were possible to verify this violation – using

sub-Planckian energies – then Susskind and Thorlacius would apparently feel obliged to

abandon black hole complementarity.

The arguments that Susskind and Thorlacius offer to dismiss these worries rest on

the geometry of the black hole spacetime and the fact that the energy required for sending

any message will be inversely proportional to the amount of time one has to send it. Let

us begin by considering the possibility of testing for violations of the prohibition on

quantum xeroxing by performing measurements on an ensemble of particles inside the

black hole and on the late-time Hawking radiation that according to the complementarian

will contain the information required to keep quantum state pure. It is clear that no signal

from the infalling observer can reach the external observer, for any such signal would

have to be superluminal. However, we might ask whether it would be possible for the

infalling observer to leave a signal inside the black hole, which the external observer

could then receive by entering the black hole himself. However, if we calculate the time

that the external observer would have to wait to recover the information from the

Hawking-like radiation, and the amount of time that the two observers would have inside

the black hole before encountering the central singularity, we find that the frequency of

the signal would have to be far beyond the Planck scale.

We reach a similar conclusion if we ask whether the external observer could ever

receive a report of the lack of a heated membrane from the infalling observer’s

perspective. If the infalling observer were to attempt to send a message between the time

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she passed through the stretched horizon and the time she passed through the event

horizon, her signal would have to use super-Planckian frequencies because the two

horizons are separated by a sub-Planckian distance. Obviously, any attempt to suspend

herself above the event horizon, or to return to the external region would also require

Planck-scale energies.

The conclusion of Susskind and Thorlacius’ argument is that any attempt to

disprove the existence of a physical stretched horizon to an observer who forever remains

outside the black hole necessarily involves energies so great that quantum gravitational

effects should come into play: “In both these experiments, efforts made to investigate the

physical nature of the stretched horizon are frustrated by our lack of knowledge of

Planck-scale physics” (Susskind and Thorlacius 1994, p. 969). From this they conclude

that our present knowledge of physics does not allow us to pose legitimately the

information loss paradox.

This claim seems to invoke a rather odd argumentative strategy that is worth

further scrutiny. Notice that we are not here ruling out the possibility of using Planckian

energies to send messages. Further, it does not seem that the objection could be that we

would not know how to generate, manipulate, or decipher Planck-scale messages. One

would assume that Morse code using Planckian gamma rays is not conceptually any more

problematic than using radio signals. Susskind and Thorlacius’ point must be that

without a full theory of quantum gravity we have no legitimate grounds to rule out their

suggestion that the seemingly incompatible effective descriptions of the infalling and

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external observers are in fact consistent effective descriptions of some underlying theory

of quantum gravity.

However, as we saw above, the argument for the applicability of standard QFT

relies on our ability to run the argument in such a way that the energies involved are low.

The fact that it would take Planck-scale energies to experimentally verify this low-energy

description seems to be irrelevant. At the very least we are owed an account of why

considerations of the energies required to verify the descriptions should be a decisive

factor in evaluating the proposals before us. Let us turn to one attempt to provide such an

account.

To begin with, let us set aside the question of the relevance of Planckian energies

and simply assume that the results of the interior and exterior measurements cannot be

known by any physical being. Does this fact have any relevance if we find

operationalism to be an untenable account of what our physical theories commit us to?

We saw above that STU argue that it will be impossible for the results of both

measurements to be communicated to any observer without generating problematic

Planck-scale energies. It is unclear, however, how this point can ground an argument for

the claim that the observables measured are complementary. STU argue that this implies

that the joint outcomes of these measurements are without “operational meaning” and that

we should therefore expect that the QFT state that encodes the correlations of these

observables are “unphysical” (p. 3744). While rejecting the unpersuasive appeal to

operationalism, we might ask whether these insights concerning the possibilities of

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communication can serve as a ground for a claim that the observables in question are

complementary.

In the cases that Bohr considered we were offered an explicit account of how an

accurate measurement of one observable prevented us from accurately measuring a

complementary observable. This was not a case of operationalism, but was an analysis of

the regime in which classical theories could reliably be employed, and the limits of that

regime. The fact that the classical regime is limited allows for the possibility of

nonclassical phenomena. The fact that the classical regime exists allows us to perform

measurements and provide a physical interpretation of the quantum formalism.

Is there a fundamental appeal to communicability in any of this that might secure

the relevance of STU’s claims that it is impossible to communicate to any physical

observer the outcomes of measurements inside the stretched horizon and measurements

on outgoing Hawking-like radiation? One fact that might catch our interest is that Bohr

and Rosenfeld’s arrangements for measuring joint field values generally required us to

connect our test bodies with rods, light signals, springs, and so on. Further, to minimize

the uncertainties of the joint measurement we had to perform momentum measurements

on the rod, discover the relative distance of the rod and one of the test bodies, etc. Only

given all of this information could we then calculate the effect of the one test body on the

other and thereby determine the original field value as accurately as possible. If this

information were in principle unavailable to any experimenter, we might feel that we had

demonstrated the presence of a new fundamental type of epistemic uncertainty, which we

might hope would open up the possibility that there might be room for a hitherto

27 Recall that to say that two regions are spacelike related means that all the points in oneregion are pairwise spacelike related to all the points in the other region – and likewisefor timelike related finite regions. I say that two regions are lightlike related just in casethere exists at least one point in one region that is lightlike related to at least one point ofthe other region. Because we are only considering simply-connected spacetimes thesethree categories are exhaustive and exclusive.

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unsuspected form of ontological indeterminacy; that is, our classical concepts might be

even more limited than we now realize. And we can assume that this sort of information

will be unavailable if one of the experiments is performed inside a black hole.

There are, however, a number of problems facing this line of reasoning. In the

case of measuring electromagnetic field values, the intermediate devices and our

knowledge of their properties and relations were only required when the behavior of one

test body had an effect on the behavior of the other. We had a clear account of the mutual

influence of the two test bodies in terms of the classical effect that they could have on

one another. By calculating the magnitude of this effect, we could arrange to compensate

for it within certain fundamental limitations. In cases where there could be no classical

influence of one body on the other – for example, when the two regions whose average

field values we are measuring are spacelike related27 – we could safely conclude that the

measurements would not interfere with one another, and therefore did not need to

introduce any compensation mechanisms or have any knowledge about the distant

measuring arrangement (aside from its spatiotemporal location). The inaccessibility of a

certain region would seem to be irrelevant if our scientific theories tell us that no

measuring arrangements in that region can have any influence on our measurement

results.

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However, this seems to be precisely the situation that we are facing in the case of

quantum fields in black hole spacetimes. The observables in question are spacelike

related, and therefore we seem to have compelling reasons to believe that one measuring

device cannot influence the functioning of the other. Thus we should legitimately be able

to conclude that both measurements can be performed, and that these measurements

reveal the true field values. It therefore seems that STU have offered us no reason to

believe that the observables in question are incompatible, and thus their appeal to

complementarity is problematic. In a moment we shall turn to the question of how

damaging this problem is for their proposal, but first we should consider one further

possibility.

One might wonder whether joint measurements, perhaps by their very definition,

would necessarily require bringing together the information revealed in the separate

measurements. If it is impossible for any physical being to possess the information of

both measurement outcomes, then one might suspect that the joint measurement of those

two observables is impossible. Perhaps this is not too far from Bohr’s picture. Recall our

discussion of measurements of average quantum electromagnetic field values represented

by noncommuting operators. Here Bohr claimed that we could measure the field value

ExII, for example, simply by compensating appropriately for the self-interaction of the test

body on itself via the field. This is also true of a measurement of ExI; all we need to do is

install an appropriately calibrated spring and we can legitimately claim to have measured

this field value to any desired degree of accuracy. However, we cannot claim to have

performed a joint measurement of ExI and Ex

II with the same degree of accuracy that we

28 This is not to say that they have not offered some motivation for this move. Forexample, STU insist that “the event space for an experiment should only containphysically measurable results” (p. 3760), and argue that the joint field values are notmeasurable. But this apparent appeal to operationalism seems illegitimate, especially inlight of the nice-slice argument that appears to offer strong support for the validity of thelocal QFT description.

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could attribute to the individual measurements considered independently. As

independent measurements we could disregard any potential interaction between the two

measuring devices – the influence of the other device would simply constitute a change in

the value that revealed itself in measurement. However, if we are performing a

joint measurement then we are responsible for any influences that one measuring device

might have on the other.

The problem once again is that the reason we cannot claim to have performed the

joint measurement in these cases is that we know that one measurement process will have

a (potential) effect on the other, and this does not seem to have been established in the

black hole case. While one might be able to build a purely epistemic case for the claim

that a joint measurement necessarily requires the ability to bring together the information

from the two measurements, this argument would take one beyond any line of reasoning

that Bohr used in developing his account of complementarity. Further, the advocates of

BHC have yet to offer a compelling account to the relevance of communicability that

would support such a position.28

Our conclusion, then, is that Susskind and his coauthors have not yet offered a

compelling case for their claim that observables inside and outside a black hole should be

considered complementary to one another – where such complementarity is to be

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understood along the lines advocated by Bohr. Of course, one might be tempted to

respond to this conclusion by simply abandoning the term “complementarity,” or by

stipulating that one has a different meaning in mind than Bohr did. The problem with a

response along these lines is that it is that there is very little left of argument for BHC if it

is stripped of its connection to the historical successes of quantum theory in the face of

seemingly contradictory results. The plausibility of BHC as a response to Hawking’s

argument for information loss rests on the claim that measurements located inside and

outside the black hole might be incompatible in a way strongly analogous to the way that

position and momentum measurements are incompatible. This analogy is then supposed

to provide room for the possibility of a unitary quantum description of the evolution of

black hole: Hawking’s argument can be rejected along the same lines that we might reject

an argument against standard quantum theory based on the claim that particles must

always be considered to have definite positions and momenta because these properties

can be measured. If the analogy between BHC and standard complementarity is weak, as

the previous arguments seem to indicate, then the plausibility of this response to the

information loss paradox is correspondingly frail. While we certainly do not have in hand

a decisive refutation of the proposal, it appears that the justification for BHC thus far rests

on an anemic argument from operationalism.

What the complementarians apparently need is some account of how a field

measurement inside a black hole can influence or disturb a measurement outside the

black hole, or vice versa. This need not be as strong as ’t Hooft’s claim that performing

one measurement would literally prevent us from performing the other measurement

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procedure. It need only be the case that we are prevented from legitimately inferring the

values in question from our measurement results; that is, the procedures would not both

count as measurements of the field values in question. One source of justification for

such a claim might come from string theory, as we shall see in Section 6.1. Another

source of justification is offered by Kiem, Verlinde, and Verlinde, to whose account we

now turn.

5.5: Space-time Complementarity

One problem faced by accounts of black hole complementarity is that of

explaining why regions of spacetime that constitute a black hole are to be treated so

differently from other regions of spacetime. An attempt to address this question is

offered by Kiem, Verlinde, and Verlinde (hereafter, KVV), who suggest in their 1995

paper that the breakdown of effective quantum field theories on classical background

spacetimes that is expected in the context of evaporating black holes, is but one

manifestation of a more general limitation of semiclassical descriptions when high

energies are involved.

According to quantum field theories in curved (classical) space time, the state of a

system, say a scalar field N, resides in a Hilbert space defined on a Cauchy slice of the

spacetime. However, the stress-energy of the field will also influence the geometry of

spacetime according to the Einstein field equations. This implies that in general different

quantum states should, in principle, be defined on differing background geometries.

Typically, however, we argue that so long as the energy-momentum of the fields we are

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studying is sufficiently small, we can safely ignore the resulting alterations of the

spacetime. This will require us to rule out any states, or interactions, that are so energetic

that their gravitational effects cannot be legitimately ignored. The limitation on the

allowable states corresponds to reducing the dimension of our Hilbert space. We can

specify these restrictions by introducing a cutoff length, ,(x), which will be the limit of

allowable field modes.

KVV suggest that the appropriate measure of when our semiclassical measure is

no longer reliable is given by the stress-energy fluctuations associated with the cutoff,

which typically grow as ,(x)!4 as the cutoff scale shrinks. Once these fluctuations reach

the same scale as the cutoff itself, the classical description of the spacetime will no longer

be accurate, and our semiclassical account will break down. However, it is not clear that

merely specifying a short-distance cutoff will offer an adequate semiclassical theory, for

any such specification seems to violate Lorentz invariance. Two different observers with

a large relative velocity will give different accounts of which wavelengths are very small

and therefore highly energetic; any choice of cutoffs would seem to privilege one of these

observers over the other. KVV’s proposal is the claim that both observers may employ

cutoff scales appropriate their particular frames, and the resulting truncated theories yield

equivalent effective quantum descriptions. The principle they offer is the following:

Space-time complementarity: A variable cutoff scale ,(x) on a Cauchy surface Gprovides a permissible semiclassical description of the second quantized Hilbertspace only when the quantum fluctuations of the local background geometryinduced by the corresponding stress-energy fluctuations do not exceed the cutoffscale itself. All critical cutoff scales that saturate this requirement providecomplete, complementary descriptions of the Hilbert space. (KVV, p. 7055)

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To say that a cutoff saturates the proposed bound is just to say that the fluctuations in

geometry induced by +TÎ :<(x)2, are the same size as ,(x). A larger cutoff would also

generate an acceptable semiclassical theory according to KVV’s proposal, but this limited

set of observables would not completely specify the Hilbert space we are looking for. To

speak of “the Hilbert space” is merely to be committed to the existence of a unitary

S-matrix that maps ingoing asymptotic pure states to outgoing asymptotic pure states.

This is not to say that the early- and late-time states will be identical Fock spaces, for the

ingoing vacuum may be mapped onto a late-time state with a nonzero particle number

expectation value; however, it is to say that the Hilbert spaces constructed using these

different cutoffs will be unitarily equivalent.

We should note that this is not what we would expect from various truncations of

a field theory. If we were to take the full Hilbert space of a QFT and discard those states

that one observer considered to be highly energetic, this description would not be

physically equivalent to a description derived by imposing a cutoff appropriate to an

observer moving very rapidly relative to the first. If the principle of space-time

complementarity is accurate, therefore, the underlying theory of quantum gravity cannot

be a true field theory. The field theoretic description gives out at energies that induce

sizable alterations of the background geometry – otherwise it would not be possible for

the various sets of low-energy observables appropriate to different observers to all be

complete sets of observables of the same Hilbert space. If KVV’s suggestion is correct,

the validity of effective field theories up to a certain energy scale will imply that above

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those scales any accurate theory cannot be defined as a field theory on a Cauchy surface

of a classically describable spacetime.

KVV argue that black hole complementarity is derivable as a special case of

space-time complementarity. The nice-slice argument discussed above demonstrates that

this claim is not trivial, for it would seem possible to pick a slicing of spacetime such that

we could describe the measurements of both an infalling observer and a late-time external

observer using the same energy cutoff – which should then imply that these spacelike

related observables commute. The central task that KVV undertake in their (1995) is a

demonstration that this is not the case. Instead they argue that their dynamical

formulation of space-time complementarity cited above implies the following kinematical

version of the principle:

Space-time complementarity (kinematical): Different microscopic observablesthat are spacelike separated on a Cauchy surface G, but have support on matterfield configurations that, when propagated back in time, have collided withmacroscopically large center of mass energies, are not simultaneously containedas commuting operators in the physical Hilbert space. Instead such operators arecomplementary. (KVV, p. 7056)

This latter principle is then directly applicable to the case of black holes for, as we have

seen earlier in this chapter, the modes supporting the observables measurable by an

infalling observer, and the modes measurable by Hawking observer, cross with very high

energies just outside the horizon.

Our derivation of Hawking radiation in Section 3.5 assumed that the ingoing

modes of the field do not interact with the outgoing modes. While the fact that we are

considering free fields seems to support this assumption, we should note that we have

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also neglected the unavoidable gravitational interaction between the modes. We can try

to calculate the effect that the stress energy carried by our field will have on the spacetime

metric, and ask how this will affect both the ingoing modes and the outgoing modes that

support Hawking radiation. A full account of the back reaction is beyond our grasp, of

course, but we can investigate some semiclassical approximations to try to develop a

heuristic account of the behavior of black holes.

KVV’s strategy is to calculate a leading order correction to the final state of the

field on a final Cauchy slice composed of future null infinity and the event horizon (i.e.,

I+chS) by deriving a gravitational interaction term that is to be added to the free field

action. The details of their calculation can be found in Appendix B. The key point that

KVV want to draw from their derivation is that the viability of a semiclassical theory

such as this one depends crucially on the cutoff chosen for the calculation. We can

impose a short-distance cutoff near the horizon that will allow us to generate a reasonable

account of the state of the field on hS. However, this cutoff will prevent us from being

able to offer a reasonable description of the outgoing state on I+, for the cutoff will have

eliminated the modes that dominated our calculation of the leading order gravitational

corrections to the outgoing states. If we wish to calculate the state on I+ we will have to

retain the high energy outgoing modes, but the presence of these modes indicates

exceedingly large gravitational interactions with our ingoing modes – which implies that

we will be unable to use our semiclassical theory to calculate the state of the field at hS

reliably. We therefore have a choice as to whether we wish to offer a reliable

semiclassical account at null infinity or at the horizon, and this choice corresponds to a

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choice in the cutoff we impose, that is, to which states we choose to eliminate from

consideration.

A third option is to limit the incoming modes near the horizon to very low

energies. This choice of cutoff length, according to KVV, will allow us to offer a

semiclassical description of the state on both I+ and hS, and the gravitational corrections

to the final state will be finite.

Qualitatively the outgoing state will look very much like the thermal state asderived by Hawking. However, it is clear that due to the necessary truncation ofthe Hilbert space the entropy associated with the black hole is drastically reducedcompared to the conventional free field result. (KVV, p. 7063)

The claim that the outgoing state will look similar to Hawking radiation seems to be

based on the observation that the corrections to the Hawking state generated by our first-

order interaction Hamiltonian will be small if our cutoff guarantees that Tvv is sufficiently

small. The entropy of the black hole that KVV are referring to is presumably to be read

off the state of the field on I+chS. This state lives in a Hilbert space ,hq,I+, where ,h

will be composed of the ingoing v-modes. When we throw out the high energy v-modes

near the horizon the dimensionality of ,h will shrink, and consequently we will have

lowered the entropy of the final state on I+ that results from discarding the portion of the

state that has fallen into the black hole.

We can now see the connection between KVV’s two formulations of space-time

complementarity. The dynamical formulation of their principle requires that a high-

energy cutoff for our effective theory not introduce stress-energy fluctuations greater than

the cutoff scale. The derivations in their paper indicate that controlling the fluctuations of

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the stress-energy tensor near the horizon of a black hole will require us to limit our

ingoing and/or outgoing modes in such a way that there will be no single choice of cutoff

that will give us an effective semiclassical theory that includes standard energy

observables corresponding to both an infalling observer and a late-time external observer.

This is the instantiation of their kinematical formulation that we are interested in.

If their calculations are reliable then KVV have shown that their dynamical

formulation of space-time complementarity implies their kinematical formulation.

However, the important conceptual question seems to be whether we have any reason to

accept their former principle. Specifically, we might be willing to accept their account of

the limitations of these effective semiclassical calculations – and the consequence that the

standard energy observables associated with infalling and external observers “are not

simultaneously contained as commuting operators in the physical Hilbert space” (KVV,

p. 7056). But how are we to justify the conclusion that these observables are

complementary?

The crucial assumption lies in KVV’s claim: “All critical cutoff scales that

saturate this requirement [that the ,(x) not introduce fluctuations larger than ,(x)] provide

complete, complementary descriptions of the Hilbert space” (p. 7055). As we noted

above, this assumption is incompatible with the high energy theory being a field theory,

and on its face is a very surprising claim. Why should we assume that these (spacelike

related) observables exist in the same Hilbert space, particularly when we have arrived at

this effective semiclassical Hilbert space only by throwing out states that support the

observables that are supposedly “complementary” to the observables safeguarded by our

29 Indeed, it appears that the outcome of two measurements that KVV considercomplementary to one another could be known by a physical being. For example, itseems that a measurement by a Rindler observer and an appropriate Minkowski observerwould be complementary to one another according to the kinematical formulation ofspace-time complementarity. However, it would seem that at the end of the measurementthe Rindler observer could stop accelerating and meet up with the Minkowski observer tocompare results. As I argue below, one should locate the relevant incompatibility in the

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cutoff? Presumably the motivation here is supposed to be based on an intuition, or hope,

that when one effective description runs out due to the geometric fluctuations implied by

the cutoff that description is based on, there will be some other effective description

based on some other cutoff that can legitimately pick up the story.

Alternatively, one might follow ’t Hooft in assuming that there should be a unitary

description of black hole evaporation, and ask what implications this assumption will

have for our effective low-energy theories. The information paradox leads us to the

conclusion that the operators associated with an infalling observer’s measurements cannot

commute with the those associated with a late-time external measurement. We may

therefore feel justified in concluding that these experiments will be represented by

noncommuting operators.

An advantage of KVV’s account is that it offers us a more plausible account of the

incompatibility between the interior and exterior measurements than we found in

positions of ’t Hooft and Susskind et al. KVV do not follow ’t Hooft in claiming that

performing a late-time measurement will destroy any infalling observer or apparatus, thus

rendering impossible any field measurement inside the black hole, nor do they appeal to

the impossibility of communicating the results of the two measurements to any

observer.29 While they do not develop a clear account of the incompatibility of the

inability of our semiclassical effective theory to account for both of these measurementsadequately.

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measurements in question, KVV seem to be taking the much more reasonable route of

arguing that even though both of these measurement procedures could be performed, they

cannot both count as measurements of the observables of interest. The relevant

characterization of the low-energy observables measured by our two observers relies on

an effective semiclassical theory requiring an appropriate cutoff length, and there is no

such theory that includes the two observables in question. We might be able to know the

outcomes of both measurements, but there would be no legitimate effective theory that

we could use to make sense of these results in an applicable low-energy semiclassical

theory. To reply to Susskind and Thorlacius’ worry of verifying information duplication,

we could argue that there is no way of knowing whether the information has been

“duplicated” in the two measurements, for there is no effective field theory that allows us

to extract the “information” from the measurement results in our hands.

KVV’s response to Hawking’s argument rests in their implicit argument that the

mere fact that all local energies on some time slice are small does not justify the validity

of the description offered by local QFT; the large virtual energies associated with

Hawking radiation can plausibly cause fluctuations in the classical geometry near the

horizon, and these fluctuations may be seen to signal a limit to the applicability of QFT.

If we insist on a more rigorous way of picking out our low energy effective theory,

according to KVV, we see that we have no compelling reason to accept the premiss of the

information loss argument that allows us to represent the late-time state of the system as a

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state operator in ,bhq,ext. However, this evasion of the information loss argument

seems to require very surprising behavior from our underlying theory of quantum gravity.

KVV’s postulation of space-time complementarity seems to offer a consistent account of

how one might evade the conclusion that information is lost in black holes, but the

plausibility of this account will rest on our whether we can offer a plausible – or even

intelligible – picture of quantum gravity that fits with these expectations of the

relationship between various effective semiclassical theories. We turn in the next chapter

to the claims that a picture of this sort may be offered by string theory.

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CHAPTER 6

EXPLANATORY POTENTIAL OF BLACK HOLE COMPLEMENTARITY

We saw in the last chapter that the work demanded of black hole complementarity

is that of providing a rebuttal to Hawking’s argument for information loss in black holes.

The core of the complementarians’ reply lies in the claim that various low-energy

effective descriptions cannot be united in the straightforward way envisioned by Hawking

and others. The arguments of ’t Hooft and STU (Susskind, Thorlacius, and Uglum) were

found to be problematic, insofar as they offered questionable resolutions of apparent

contradictions between the descriptions offered by different observers. The position of

KVV (Kiem, Verlinde, and Verlinde) seemed more promising, in that they at least

postulate a theoretical structure that would resolve the apparent contraction, but it still

suffers from being extremely speculative. A substantial justification of black hole

complementarity would seem to require a full theory of quantum gravity that would allow

us recover the effective theories in question, and would also allow us to analyze the

relationships between the “descriptions” offered by various observers.

In Section 6.1 we shall investigate the claim that string theory promises to fill this

role. The strongest support for the unitary evolution of black holes comes from the

behavior of models that purport to offer examples of such evolution. However, such

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results are at best preliminary at this point. Section 6.2 returns to Bohr’s account of

complementarity and begins the work of evaluating the explanatory force of his account.

Key assumptions in Bohr’s argument will be highlighted by comparing his position to a

hidden-variables interpretation of quantum mechanics. We conclude in Section 6.3 by

investigating the costs and benefits of accepting complementarity as a resolution of

paradoxes presented by modern physics.

6.1: String Theory and the Holographic Principle

As we have seen, BHC is primarily an account of the relationship of effective low-

energy descriptions appropriate to different observers – or better, appropriate to certain

frames of reference. Even if the suggestion of the complementarians is correct, we would

ultimately like to know what the full theory of quantum gravity underlying these effective

descriptions is. Indeed, it may be that until we have such a theory in hand we will have

no way of judging whether their description of the relationship between various effective

theories is correct.

While ’t Hooft (1987, 1990) was apparently the first to suggest that there was a

fundamental connection between the evolution of black holes and the behavior of

superstrings, Susskind became the chief proponent of the claim that string theory exhibits

the sort of behavior predicted by black hole complementarity. This claim is largely based

on three potentially related features of string theory. The first is that strings seem to

spread spatially with increased temporal resolution. The second is the rather speculative

suggestion that the number of degrees of freedom in the fundamental theory of quantum

30The argument here is over the question of what should be required of a “genuine” theoryof gravity. String theory requires a background spacetime. Because GR is a theoryof spacetime some argue that it is illegitimate for a theory of QG to help itself to somedynamically independent background spacetime metric. The interested reader is referredto the collection by Callender and Huggett (2001), which includes a number of articlestouching on this issue.

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gravity, which is presumed to be an extension of string theory, will scale as the area of the

boundary of a region, and not as the volume of the region. The third feature is the

potential failure of microcausality over large distances: it is argued that one can use string

field theory to derive a nonvanishing value for the commutator of operators representing

spacelike related observables in a black hole background spacetime. As we shall see,

each of these suggested features seems to support the picture of black hole evolution

advocated by the complementarians.

While the details of string theory are beyond the scope of this dissertation, it will

be helpful to have a rough picture of the nature of this theory. Instead of starting with a

point particle, whose action is defined along a world line, this theory considers an

extended string, whose action will be defined along a two-dimensional “world sheet,” and

whose classical equations of motion will be given by varying that action to find its

stationary points. The consistency of this theory requires the incorporation of a symmetry

between bosons and fermions called supersymmetry; thus the theory is often referred to as

superstring theory. The quantum theory of these strings necessarily includes spin-two

particles, which are identified with gravitons, the carrier of the gravitational force.

Therefore, string theory is arguably a theory quantum gravity.30

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A feature of this theory that is important for our purposes here is the claim that the

size of strings depends on the resolution time, ,, which is the period of time that a

particular measurement is sensitive to. Susskind (1970) showed that as this resolution

size shrinks, the size of the string in the transverse direction grows as

R2string ~ ln(1/,), (6.1)

but the consequences of this fact for descriptions of the formation and evaporation of

black holes were not explored until Susskind’s (1993). Here he points out that an

external fiducial observer making observations on an object falling into a black hole will

see all physical processes slow down as the object nears the horizon. This time dilation

effect implies that measurements performed by an external observer will be probing very

high frequencies of the string, effectively employing a resolution time of

, ~ exp(-t/4M),

where t is Schwarzschild time, and M is the mass of the black hole. This implies that

experiments performed by the distant observer will reveal a spreading of the string as it

nears the horizon.

The string will also spread in the longitudinal direction, eventually counteracting

the Lorentz contraction that an external observer will see as the string falls toward the

horizon. The picture we are led to is that “someone observing an infalling string will find

that it appears to grow and cover the horizon forming a sort of stringy goo just above the

horizon” (Susskind and Uglum 1996, p.127). This scenario fits quite nicely, of course,

with the suggestion that an external observer can legitimately treat a black hole as a

quantum mechanical heated membrane sitting at the stretched horizon. Infalling

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observers, however, will notice no such effects, because they will not be able to probe

this high energy behavior. We should therefore not expect such observers to observe a

heated membrane, or anything out of the ordinary at the horizon.

While this picture fits the account of BHC very well, its argumentative force will

depend significantly on what exactly is involved with the “spreading” of strings described

above. Susskind (1994) clarifies the issue by arguing that the central question here is the

size of the area over which the information about the state is spread. We can consider a

screen of minute detectors, such as a photographic plate, with which a particle or a string

collides. If we investigate the size of the mark the object leaves on this plate, we find that

it will grow with the energy of the particle. Susskind argues that this is true of both

particles – i.e., in local QFT – and of strings. However, in the case of particles, the area

over which the information carried by the state is spread remains bounded as the energy

of the particle increases, even though the size of the spot itself continues to grow. If we

consider two different particles that strike our screen of detectors, in principle we should

be able do identify each of them based on the response of the detectors. We can then ask

about the area of the detection screen that is relevant to distinguishing between these two

states. Susskind argues that while in particle physics this area is generally bounded, in

string theory it grows as Equation (6.1).

The string size under consideration is thus a measure of the possible interactions

between the string and other systems. According to a late-time external observer an

infalling string can interact in a significant (information-transferring) way with the entire

horizon – while an infalling observer will observe no such interactions. Contradictions

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obviously threaten here, for it is difficult, if not impossible, to reconcile a claim that an

object – whether a string or an observer – interacts with many other systems at the event

horizon, with a claim that this same object did not, and could not, interact with these

other systems. The tension between these two claims grows when we recall that

according to the external observer’s account these interactions will destroy the infalling

observer. A resolution of this tension would seem to require a recognition that there is no

legitimate description of the scenario that allows the infalling observer to survive for all

time: even the infalling observer admits to her own destruction when she nears the center

of the black hole. It therefore seems that it would be a mistake to characterize the

difference between the internal and external observer’s accounts as resting in

whether infalling objects violently interact with the systems that compose the black hole:

the difference instead lies in the variety of the descriptions of when and where these

interactions occur.

The suggestion that the fundamental theory of quantum gravity may require some

revisions of our concept of spatiotemporal localization is echoed by the so-called

holographic principle. This speculative principle can be viewed as a natural conclusion

of taking black hole thermodynamics seriously. Recall from Chapter 3 that the parallel

between black hole mechanics and the laws of ordinary thermodynamics seemed to imply

that a black hole could be assigned an entropy proportional to its surface area. According

to our usual interpretation of entropy, this should imply that the black hole can be

assigned a number of accessible states given by the exponential of this entropy. It has

been a goal of many who take black hole thermodynamics seriously to find an account of

31 It is worth noting that these dualities generally connect different perturbative stringtheories, and thus are taken to show that what were previously taken to be distincttheories are actually different facets of a single fundamental theory that has yet to bediscovered. An introduction to these issues can be found, for example, in Polchinski(1998a, 1998b).

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these microstates underlying the thermodynamic description of the black hole. A 1996

paper by Strominger and Vafa provides such an account for a certain class of black holes.

Strominger and Vafa’s derivation was made possible by the discovery in string

theory of a new set of theoretical entities: extended objects referred to as “branes” (from

“membranes,” which are a two dimensional branes). So-called “D-branes” (for “Dirichlet

branes”) are subspaces that mark the ends of open strings whose coordinates normal to

the brane obey Dirichlet boundary conditions. These D-branes are dynamical entities in

their own right and are intimately related to the fundamental objects of the theory, i.e, to

the strings. For example, we can begin with D1-brane (i.e., a one dimensional D-brane,

or a “D-string”) in a perturbative string theory with a weak coupling constant g, and

consider what happens as we increase the coupling constant. In the large g limit we find

that this entity behaves like a fundamental string (“F-string”); that is, we recover the

dynamics of an ordinary string. We then say that the D-string is dual to the F-string.31

The study of the transformation of branes under dualities of this sort revealed that

it is possible to find black hole solutions to classical field equations that will be dual to a

collection of objects in a string theory. More specifically, working in superstring theory

in 9+1 dimensions, we can consider a “black p-brane,” which is extended in p

dimensions, but which has a black hole geometry in the other 9!p dimensions. Under a

transformation of the coupling constant we can show that this object will be dual to a D-

32This fifth dimension is taken to be large (i.e., it is not compactified), but finite. Thuswhat I am calling p5 should be an integer value given by Lp~5/2B, where L is the length ofthe fifth dimension and p~5 is the continuous momentum.

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brane, or a collection of D-branes. We generally cannot follow such a transformation,

because it takes us into a region where the coupling constant is too large to use

perturbation theory. However, there will be a certain class of D-branes and black p-

branes for which supersymmetry will guarantee that even though we are passing through

a nonperturbative region of our theory, the state counting that we perform on the D-

branes will have to remain valid as we pass over to the black p-brane.

Strominger and Vafa’s aim was to construct a supersymmetric black hole with a

smooth horizon and a nonzero area, so that the black hole would have a definite

Hawking-Bekenstein entropy that might be mirrored by the string-theoretic construction.

Such a black hole requires at least three charges, which we will refer to as Q1, Q2, and p5,

where this last is the momentum in the fifth dimension.32 We consider a ten-dimensional

spacetime that has the sixth through ninth dimensions compactified and which contains

an extremally charged black hole, that is, a black hole that does not Hawking radiate

because its charges precisely counterbalance its gravitation. The charges of such a black

hole determine its surface area and thus its Hawing-Bekenstein entropy. If we take the

charges and area to be large, so we are well away from the Planck length and have a

classical black hole, the entropy of this hole is given by (Polchinski 1998b, p. 221)

SBH = 2B(Q1 Q2 p5)1/2.

We now can consider this same black hole as a collection of D-branes that carry

the appropriate charges dual to Q1 and Q2. These will be D1-branes in the 5-direction,

33It is worth noting that the loop quantum gravity program has yielded a derivation of theBekenstein entropy of a black hole (and this has been counted as a success of the programby its proponents). However, there is a parameter in loop quantum gravity that must beproperly tuned for the derivation to match the Bekenstein result (though once it is chosenfor a single black hole, the same parameter setting yields the correct result for all otherblack holes), while string theory offers the precise result without any fine tuning.

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and D5-branes in the (5,6,7,8,9)-directions. An investigation of the partition function

describing a bound state of these branes reveals the following density of states

(Polchinski 1998b, p. 222):

N(E) = exp[2B((Q1 Q2 p5)1/2],

where E is the energy of the state (given by 2Bp5/L). The fact that the entropy of these

branes matches that of the dual black hole solution has been taken by many high energy

physicists to be a very strong indication that string theory offers an account of the correct

quantum-gravitational degrees of freedom. Although Strominger and Vafa’s original

state counting argument concerned a very specific five-dimensional extremal black hole,

it has since been extended to black holes of other dimensions – including four dimensions

– and black holes that are not precisely extremal. One encounters limits when one tries to

consider more realistic black holes, such as Schwarzschild black holes, because without

sufficient symmetries to connect the descriptions of the states in the two regimes one

cannot neglect the nonperturbative behavior of the theory. One can recover the correct

order of magnitude for Schwarzschild black holes, but the numerical coefficient cannot be

derived.33

We now turn to the question of what implications this state counting argument has

for the information loss debate. Most string theorists take this to be a strong argument

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against the possibility of nonunitary evolution of black holes. We now seem to have

evidence that black holes are really string, or D-brane, states, and, as far as we know,

these entities evolve strictly unitarily. However, this argument is not fully compelling,

even to the string theorists, because our current understanding of string theory is largely

based on perturbation theory, and the black hole state-matching argument necessarily

takes us through regions of strong coupling where these techniques cannot be applied.

The fact that the weak-coupling perturbative theories are unitary need not imply that the

“full” theory – now generally referred to as M-theory – is also strictly unitary, but it does

weaken the plausibility of nonunitary evolution.

This derivation has also given support to the belief that the Bekenstein entropy is

an accurate measure of the number of fundamental degrees of freedom underlying a black

hole. ’t Hooft was the first to emphasize that this seems to imply the radical conclusion

that the maximum number of degrees of freedom in any region of space will be

proportional to the square of the radius of that region, and not the cube of that radius as

our current theories would lead us to believe. Suppose that a system in a given region of

space has number of accessible states greater than the N = exp(SBH) that would be

assigned to a black hole filling that same region. We could now add matter to that region

until it formed a black hole – and apparently our adding matter to the region would result

in fewer fundamental degrees of freedom than were originally present. This possibility

seems objectionable on a number of grounds including the fact that it would violate the

generalized second law of black hole thermodynamics.

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The apparent contradiction here has led ’t Hooft and others to postulate that the

Bekenstein entropy gives a maximum bound on the number of degrees of freedom in any

region. As was the case with black hole complementarity, the popularity of ’t Hooft’s idea

grew after it was championed by Susskind and given a catchy name, in this case, the

“holographic principle.” The essence of this principle is the claim that the maximum

number of degrees of freedom in any finite volume of spacetime is finite – and this

number increases as the area, and not the volume, of that region.

A little reflection should convince the reader that this suggestion is truly

counterintuitive. Any standard field or particle ontology will lead us to conclude that if

we have some region O composed of various subregions oi, and each subregion can be

assigned some number of degrees of freedom ni, then the total number of degrees of

freedom, N, should be at least as large as the sum of the ni, i.e.,

N $ Ei ni.

However, according to the holographic principle, N # Ei ni, because surface area only

goes up as r2 whereas volume, along with the number i of subregions in O, goes as r3.

The picture implied by this principle is that, in some sense, everything within a

volume can be mapped without residue onto a grid on the surface of that volume, where

each Planck area square of the surface will be assigned a single binary degree of freedom.

This principle is little more than a conjecture at this point, but it has received some

noteworthy support from the discovery of the so-called AdS/CFT correspondence. This

is a correspondence between string theory defined in anti-deSitter spacetime (AdS) and a

conformal field theory (CFT) defined on the boundary of that spacetime. One of the

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surprising features of this correspondence is that the string theory in the “bulk” – i.e., in

the full spacetime – includes gravity, but the CFT on the boundary has no explicit

gravitational degrees of freedom. The fact that the gravitational processes in AdS can

apparently include evaporating black holes, together with the fact that the corresponding

CFT description of the process is completely unitary, lends support to the claim that

information is not lost in black holes. Furthermore, this correspondence goes some way

towards offering an example of holography – though it is not a full realization of the

principle because it does not include a limit to a single Boolean degree of freedom for

each Planck area on the border.

The details of these developments in string theory go beyond the scope of this

dissertation, but a fair evaluation of black hole complementarity requires us to take note

of the sort of physics that its proponents hope to find to undergird their position. We

should also note that this view is not without its critics. Wald, for example, argues that

the above argument for the holographic bound might fail even if the generalized second

law holds. He suggests that it might turn out to be impossible to form a black hole using

a system with a number of degrees of freedom in excess of the holographic bound, for the

mass we add to the system in hopes of creating a black hole may Hawking radiate away in

less time than would be required for gravitational collapse to occur (2001, p. 23). Wald

also points out that the holographic bound (like the Bekenstein bound discussed in

Section 3.5) would be violated by having a sufficiently large number of particle species.

In trying to flesh out the relationship between holography and BHC, it is helpful

to notice that both are incompatible with a fundamental local field theory – and the

34Susskind has argued in private communication that such violations of microcausality donot threaten to lead to causal paradoxes because the analyticity of all superstringscattering amplitudes implies that one would not be able to use such noncommutativity tosignal. It is standard to argue that causality implies the analyticity of the S-matrix, but itis less well known that the implication is a biconditional. See, for example, Byron andFuller (1992), p. 347.

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violation of locality suggested by both scenarios seems to require intricate relationships

between various low-energy descriptions. The nonlocality cannot be restricted to some

fundamental distance scale, such as the Planck length, but must be able to manifest itself

over arbitrarily great distances, for example, over the entire horizon of an arbitrarily large

black hole. This nonlocality will likely manifest itself in a lack of commutativity between

some operators that would naturally be associated with low-energy spacelike related

observables.34 For example, we have seen that the nice-slice argument for information

loss rests only on the assumption that microcausality holds to a good approximation for

all local observables that have low locally-defined energies on some spacelike time slice.

Susskind and Thorlacius – as well as Kiem, Verlinde, and Verlinde – tried to undermine

this argument by pointing out that it appeals to a potentially controversial assumption

about how the locality of low-energy physics will be coded into the full theory of

quantum gravity. Kiem, Verlinde, and Verlinde claimed that the relevant criterion for a

low-energy semiclassical theory is the size of the fluctuations in the metric implied by the

cutoff used by the theory. Pairs of observables that did not admit of a common suitable

cutoff – such as those localized inside a black hole and at I+ – were claimed to be

complementary, meaning that their corresponding operators should fail to commute.

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In 1995, Lowe, Polchinski, Susskind, Thorlacius, and Uglum bolstered this line of

argument by showing that the nice-slice argument fails in light-front string field theory.

They consider a series of nice slices in a black hole spacetime of arbitrarily large mass –

i.e., Rindler spacetime – and consider the commutator between operators defined on these

nice slices. They are able to show that the commutator between operators representing

observables localized inside and outside the black hole fails to vanish, even though these

are spacelike related observables that have low nice-slice energies.

It is worth noting that the status of string field theory is somewhat controversial

even in the string community. This is partly because the theory assumes that the degrees

of freedom given by a particular perturbative string theory accurately capture the behavior

of the full theory, and the belief that perturbative string theories are merely limits of some

more fundamental theory – viz., M-theory – calls this extrapolation into question.

Nonetheless, the argument put forward by Lowe et al. does at least call into question the

assumption that we can safely argue from the fact that the low-energy limit of a nonlocal

theory such as string theory is a local quantum field theory, to the claim that

microcausality will be obeyed whenever such a theory can be defined. The fault in this

argument, according to Lowe et al., lies in the fact that even though the entire system has

a low nice-slice energy, the extreme kinematic disparities over the nice slice will imply

that commutators between distant field operators can involve high energies, and thus can

exhibit nonlocal behavior – i.e., they need not vanish even though they are spacelike

related.

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While we cannot pursue the details and implications of this proposed nonlocality

here, we shall want to ask whether complementarity is playing any significant role in this

speculative account of quantum gravity and its relationship to more familiar theories. To

set the stage for this discussion, let us return to our discussion of Bohr’s complementarity.

6.2: Classical Concepts and Hidden Variables

Chapter 1 focused on explicating Bohr’s account of complementarity so that we

might judge whether BHC should be viewed as employing this account to resolve the

paradox posed by Hawking. In Chapter 5 we saw that extant arguments for BHC lack an

ingredient that is a necessary part of a legitimate appeal to complementarity, namely, a

rigorous account of the incompatibility of the contexts that allow us to apply our

complementary descriptions. We have not argued, however, that such an account could

not be developed. By way of conclusion, it might therefore be worthwhile to offer an

evaluation of complementarity as an explanatory and interpretive strategy in resolving

difficulties facing our physical descriptions of the world. To this end, let us ask whether

Bohr offers us compelling reasons for believing that any adequate account of quantum

mechanics will require an appeal to complementarity.

Recall from Section 1.5 that Bohr believes that our classical concepts such as

position and momentum face fundamental limitations, and these are limitations of the

concepts themselves and not merely of our ability to measure the properties they describe.

He believes that this follows from the fact that if classical concepts were universally

valid, then quantum mechanics – with its nonclassical phenomena, such as stationary

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states, particle interference patterns, etc. – would be impossible. Thus nature herself has

taught us that there are fundamental limitations to the validity of our classical

descriptions.

This is not, according to Bohr, merely one possible conclusion to draw from the

empirical success of quantum theory. It is a conclusion that is forced on us.

[I]n quantum mechanics, we are not dealing with an arbitrary renunciation of amore detailed analysis of atomic phenomena, but with a recognition that such ananalysis is in principle excluded. The peculiar individuality of quantum effectspresents us, as regards the comprehension of well-defined evidence, with a novelsituation unforseen in classical physics and irreconcilable with conventional ideassuited for our orientation and adjustment to ordinary experience. (Bohr 1949, p.235; BCW 7:375, Bohr’s emphasis)

However, Bohr is mistaken here. There is, as we shall see momentarily, a perfectly

consistent interpretation of the quantum formalism that attributes definite positions and

momenta to all particles at all times. Before turning to this hidden variable theory,

however, it will be helpful to try to get a better sense of why Bohr believes that such a

theory is impossible.

Planck’s constant is, of course, the essentially new nonclassical element in

quantum mechanics. Where does this constant enter our theory? Bohr tells us,

In this formalism, the canonical equations of classical mechanicsdpi/dt = !MH/Mqi , dqi/dt = !MH/Mpi

are maintained unaltered, and the quantum of action is only introduced in the so-called commutation rules

pi qi ! qi pi = h/2Bifor any pair of canonically conjugate variables. . . . In particular is the essentiallystatistical nature of this account a direct consequence of the fact that thecommutation rules prevent us to identify at any instant more than a half of thesymbols representing the canonical variables with definite values of thecorresponding classical quantities. (Bohr 1939, p. 14-15; BCW 7:306-307)

35A nice, thorough exposition of Bohm’s theory is offered in Holland (1993). SeeCushing 1994a and 1994b for a discussion of how Bohm’s theory answers Bohr’sarguments for complementarity and for the historical genesis and reception of this view.

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In quantum mechanics we retain the classical Hamiltonian, or a formally equivalent

analogue of it. If the value of the generalized coordinates and momenta were also

classically well defined, even if unknown, then all behavior would be classical. But

clearly this is not the case, for classical mechanics cannot account for the black-body

spectrum, stationary states, particle interference patterns, etc. Thus, Bohr’s reasoning

seems to run, the existence of these phenomena implies a limitation on the extent to

which we can identify the quantum “symbols” pi and qi with “definite values of the

corresponding classical quantities.”

Such a line of argument hardly constitutes a proof of the impossibility of

supplementing quantum theory with hidden variables. We might wonder whether it

would be possible to modify, or abandon, the standard Hamiltonian formalism in such a

way that the statistical predictions of quantum mechanics would be unaltered, but the

values of both position and momentum would be well defined. We might also consider

whether there is a more subtle way of interpreting the quantum operators p$x and x$ (Bohr’s

quantum “symbols” pi and qi) in terms of classical values – or equivalently, of recovering

the values of the classical properties px and x from our theory – than the straightforward

identification that Bohr adopts.

Indeed, David Bohm developed just such a theory in 1952.35 An examination of

how this hidden-variable theory avoids Bohr’s arguments, and how it attributes values on

the basis of measurements, will help to clarify the work that Bohr’s account promises to

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accomplish, and the difficulties that we face in confronting the limits of theoretical

descriptions. Bohm supplements the quantum formalism with precise positions for all

particles. The momentum of each particle is given by

p = LS,

where S is the phase of the wave function given by the decomposition of the complex

function R(x) into two real functions R(x) and S(x):

R = ReiS.

In this theory all particles have completely well-defined positions and momenta at all

times; however, our ignorance of the precise positions of the particles involved will

prevent us from being able to predict the outcomes of measurements more precisely than

is allowed by Bohr’s interpretation. The time evolution of the particles will be given by

the modified Hamilton-Jacobi equation:

MS/Mt + (LS)2/2m + V + Q = 0, (6.2)

where V is the classical potential and Q is the so-called “quantum potential” defined by:

Q = ! S2 LR / 2m R. (6.3)

It is this quantum potential that can be seen as the source of all “non-classical” quantum

behavior. Because the quantum potential need not fall off at large distances, Bohm’s

theory is highly nonlocal in general.

From Equation (6.2) and the prescription that p = LS, or equivalently v = LS/m,

we can establish a vector field that will fix the trajectories of the Bohmian particles.

These trajectories depend on the overall wave function, as can be seen in Equations (6.2)

and (6.3), and thus can be instantly altered by changes in the distant part of the wave

36This suggestion has been raised by Don Howard (private communication).

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function. However, it can be shown that these instantaneous actions at a distance will be

undetectable if we assume with Bohm that the initial distribution of particle positions of

an ensemble of Bohmian states is given by |R(x)|2.

Bohm’s theory is completely deterministic. Given the wave function for the

system and the initial positions of the particles, the trajectory of the particles are

completely fixed by Equation (6.2). The only uncertainty in the theory comes from our

inability to discover the precise initial positions of the particles. Generally this inability is

due to the fact that a measurement will disturb the state of the system we are observing

(see, e.g., Holland 1993, Ch. 8). This, by the way, is one of the more intuitive ways of

seeing the shortfalls of the naive disturbance interpretation of QM: One can assume that

at all times quantum systems have definite properties, which are disturbed by

measurement, but then one will be left with Bohm’s interpretation, or some related

theory. This comes with a certain price that most naive disturbance theorists are unaware

of, namely nonlocality.

This violation of locality, together with the apparent tension that it creates for

reconciling QM with relativity theory, is one of the chief objections that Bohm’s theory

faces. While we cannot pursue this issue here, it is worth asking whether this is an

objection that Bohr might have had in the back of his mind when objecting to hidden

variable theories.36 One can find some vague hints to support this suggestion; consider,

for example, one of Bohr’s explanations of the significance of our inability to establish

37For more details of this argument, see Holland (1993), Ch. 8, which I largely followhere.

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which slit the particle passes through in a two-slit experiment if we wish to preserve the

interference pattern:

This point is of great logical consequence, since it is only the circumstance thatwe are presented with a choice of either tracing the path of a particle or observinginterference effects, which allows us to escape from the paradoxical necessity ofconcluding that the behaviour of an electron or a photon should depend on thepresence of a slit in the diaphragm through with it could be proved not to pass. (Bohr 1949, p. 217-218; BCW 7:258)

This “paradoxical” consequence is, of course, brazenly accepted by Bohm’s

interpretation. However, we might wonder whether it is reasonable to criticize Bohm’s

theory on this point if the alternative is to abandon spatial descriptions except in the

context of a measurement designed to record the position of a particle. Is non-spatiality

really preferable to well-defined non-locality? This debate is not one we need to pursue

here, however, for I wish to argue that it is a different worry that Bohr most likely had in

mind when ruling out the possibility of hidden variable theories.

This worry concerns the way that hidden variable theories such as Bohm’s go

about defining physical values such as momentum and how these definitions are

connected to measurement. Bohm defined momentum as the gradient of the phase of the

wave function (p=LS). This implies that standard “momentum measurements” will

generally not reveal the preexisting value of momentum – the measurement outcome will

instead be an artifact of the measuring process. Only when the original state of the

particle is a momentum eigenstate, given by eipx, will the measurement result, p, coincide

with the actual value of momentum given by LS. This can be seen as follows.37

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Before the interaction between the particle and the apparatus the total state of the

system is given by the factorizable wave function

Q0(x, y) = R0(x) N0(y).

An ideal “momentum measurement,” as discussed in Section 1.5, will involve an

interaction Hamiltonian that will couple the momentum, p$x, of the object to the conjugate

momentum, p$y, of the “pointer needle” of the apparatus:

H$ I = k p$x p$y.

Under the evolution of this Hamiltonian, an initial wave function for the combined

system Q(x, y, t0) will evolve according to the Schrödinger equation:

iS MQ/Mt = H$ I Q = !k p$x iS MQ/My. (6.4)

To solve this, we expand Q in terms of x-momentum eigenfunctions:

Q(x, y, t) = (2BS)-½ Idpx M(px, y, t) eipx.

Using Equation (6.4) we see that M(px, y, t) will evolve as

MM/Mt = !k p$x MM/My.

If T is the duration of the interaction, and M0(px, y) is the initial x-momentum distribution

of our state, then this yields

M(px, y, T) = M0(px, y ! k px T),

where the right hand side does not explicitly depend on T because all the time

dependence is contained in the change in y. We can expand our initial wave function

R0(x) as

R0(x) = (2BS)-½ I dpx n(px) eipx/S,

which gives us

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M0(px, y) = n(px) N0(y).

Thus, our total wave function after the interaction is given by

Q(x, y, T) = (2BS)-½ Idpx n(px) eipx/S N0(y ! k px T).

If the coupling constant, k, is strong enough, and the duration, T, is long enough, then the

partial wave function of the particle, R(x), will become a “momentum” eigenstate, eipx/S.

Thus the actual initial momentum of particle, given by LS, or equivalently

Im(R†LR/R†R), will be changed by the measurement interaction unless R0(x) is an

eigenfunction of the momentum operator, p$x. If the wave function is not such an

eigenstate then the particle’s momentum will depend on its precise location. The gradient

of the phase generates a momentum field that will generally vary from position to

position. Therefore particles with identical quantum wave functions will generally have

different momenta as a consequence of their different positions. Further, the actual

energy-momentum of Bohmian particles will not be conserved – although this fact will

again be masked by the quantum potential and the fact that the probability distribution of

the location of the particles is given by |R(x)|2.

Let us now compare this situation with Bohr’s dictum that an application of a

concept such as momentum requires us to produce a measurement arrangement that will

allow us to use classical physics to infer the property value of object from some value of

the apparatus. Recall that according to Bohr this is forced on us by the very meaning of

the word ‘measurement’:

[A] measurement can mean nothing else than the unambiguous comparison ofsome property of the object under investigation with a corresponding property ofanother system, serving as a measuring instrument, and for which this property is

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directly determinable according to its definition in everyday language in theterminology of classical physics. (Bohr 1939, p. 19)

One of the key conceptual moves that allows Bohm to escape Bohr's argument is that in

his hidden variable theory "momentum measurements" do not generally measure (the

true) momentum of the particle. It is crucial for Bohr's account that the way we attribute

property values of classical properties (such as momentum) to objects (and, implicitly, the

only way we legitimately attribute such values) is to read it off of a measuring device that

is specifically constructed to correlate a pointer needle position – i.e., some observable

degree of freedom – with the appropriate "property" of the quantum phenomenon. One is

not forced, however, to accept Bohr's account of the use of measurements to attribute

properties to objects. Bohm demonstrated that we have other options at our disposal: We

can adopt a theory that reinterprets phenomena that were previously considered to be

accurately described by our classical theories.

Bohr apparently never commented on Bohm’s theory, so the exercise of

determining how he would reconcile his arguments for the inevitability of

complementarity with the existence of an empirically adequate hidden variables theory

requires some degree of speculation. Fortunately, however, we do have the comments of

his disciple Rosenfeld. Of course, one needs to exercise caution in attributing a

spokesperson’s views to Bohr – but in this case Rosenfeld’s criticisms are entirely

consistent with Bohr’s position.

Bohm’s argument is very cleverly contrived. One would look in vain for anyweakness in its formal construction. What a paradox! Here is a faithfultranslation of all the formulae of quantum mechanics into a language which to allappearances is that of classical mechanics. . . . . Yet, all this seductive

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construction is just a sham. It is Bohm’s pleasure to give his ‘hidden parameters’such names as coordinate and momentum, but it is a far cry from the name to thething. In order to be sure that such and such a parameter really represents theposition of a particle it is necessary to examine its relation to the spatial system ofreference of some observer, in other words to analyse the measurement of theposition. But then, as one would expect and as Bohm conscientiously proves,one finds that the identification of the parameters with the corresponding physicalconcepts is only justified within the limits of the uncertainty relations. Thus, inthe end, this subtle and laborious circuit leads us back again to complementarity. (Rosenfeld 1953, pp. 402-403)

If Rosenfeld were more familiar with the theory he might have realized that his example

of a position measurement is inappropriate, for this is one case where Bohm’s theory tells

us that an ideal measurement actually does reveal the pre-measurement value of the

particle’s property. A position measurement actually measures position, even by Bohr’s

standards. If anything deserves the label “sham” it is the relationship between the particle

momentum postulated by the theory, and the “momentum” revealed in ideal

measurements – for here indeed Bohm’s theory does not accord with Bohr’s analysis of

the proper attribution of classical concepts. Nonetheless, Rosenfeld’s instance that we

must “analyse a measurement” (emphasis Rosenfeld’s) before we can attribute a meaning

to a parameter is a faithful reiteration of Bohr’s dictum that “experimental conditions . . .

constitute in fact the only basis for the definition of the concepts by which the

phenomenon is described” (Bohr 1939, p. 24; BCW 7:316); and, if my reading of Bohr is

correct, it is precisely this principle that is violated by hidden variable theories such as

Bohm’s.

We should not think that Bohr was simply so attached to classical physics that he

was unable to conceive of abandoning the concepts and definitions it offered. He did, of

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course, place very heavy emphasis on his correspondence principle, which would not

naturally lead one to define the momentum of a quantum particle as the gradient of the

phase of its wave function. But in developing quantum theory he was quite prepared to

abandon or alter either the classical ideas of energy-momentum or of spacetime, if this

would yield an adequate model of the workings of an atom and the interactions between

light and matter. Indeed, in his 1924 paper with Kramers and Slater, he suggested that the

concepts of energy and momentum could be divorced from strict conservation laws, and

that such laws should be treated as only statistically valid. However, the strict validity of

the conservation of energy-momentum in individual interactions was demonstrated –

decisively, in Bohr’s opinion – by experiments by Bothe and Geiger.

However, the discrete, particulate nature of light required for energy-momentum

conservation is nearly impossible to reconcile with the field nature of classical

electromagnetism. The paradox that Bohr faced was not the failure of spatiotemporal

and/or energy-momentum descriptions – but rather their validity even in the quantum

realm. When we perform a measurement to test whether or not energy-momentum is

conserved, we find that it is. But spatial descriptions are also indispensable. How else

are we to explain interference phenomena, which clearly depend on the spatial relations

between the slits and the screen? Bohr’s problem is that both of these descriptions seem

to face no intrinsic limitations on their validity – and yet we cannot accept them both and

still account for stationary states, the black body spectrum, the building up of interference

patterns from the registration spots of single particles, and so on.

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Bohr finds his solution in 1927. The limitations that these concepts face are not

intrinsic limitations, but are rather limitations that can be reciprocally made more or less

acute. The limitations that are in place will depend on the measurement context that we

have arranged. In discussing the conflict between optical effects, such as interference,

and the conservation of energy-momentum that is demonstrated in the photoelectric and

Compton effect, Bohr emphasizes the fact that both of these aspects of classical

description have been experimentally verified.

As is well known, the doubts regarding the validity of the superposition principleon the one hand and of the conservation laws on the other, which were suggestedby this apparent contradiction, have been definitely disproved through directexperiments. (Bohr 1928, p. 580; BCW 6:148)

Note that we have definite proof through direct experiments that the superposition

principle is valid, and that the conservation laws are strictly valid. These are apparently

facts about the world that nature herself has taught us – our job is to accept them and find

some way of resolving apparent contradiction that they generate. The solution, of course,

is complementarity – an account of the limits that these undeniably valid descriptive

frameworks face.

This, I believe, is the heart of Bohr’s belief in the inevitability of

complementarity. The strict conservation of energy-momentum leads him to attach the

very concept of momentum to these laws. Momentum just is that quantity that is

conserved in all collisions. This allows him to offer an account of when this concept can

“unambiguously” be applied – namely when we have correlated the property of the object

we are interested with some readily observable property using the laws of classical

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physics in their proper domain of applicability. A similar prescription holds for the

application of spacetime coordinates. The beauty of the theory, or interpretation, lies in

the fact that these conceptual limits dovetail perfectly, as is evidenced by the quantum

formalism.

As Bohm’s theory has demonstrated, complementarity is not necessitated by the

empirical adequacy of nonrelativistic quantum mechanics, but it may nonetheless be a

consistent, and perhaps attractive, interpretation of the quantum formalism. In our final

section we look at some strengths and weaknesses of the viewpoint of complementarity

and draw out the implications of this discussion for attempts to interpret our physical

theories.

6.3: Complementarity and the Limits of Descriptions

It is a plain truth that most representations are limited. The visual horizon that

hides the setting sun and the far side of the Earth is an obvious example of the limited

ability of a single viewpoint to capture reality. We can change our perspective to include

other features of the world, but only at the price of hiding some features that are presently

visible. Similarly, as we saw in Chapter 3, the Schwarzschild solution to the Einstein

field equations can be seen as a limited coordinate-based description of a fuller spacetime

that includes the region inside the event horizon. We generally find examples such as

these untroubling, for there are choices of coordinates that will allow us to extend our

general relativistic description to the interior of a black hole, and one can in principle

offer a description of the entire surface of the Earth. Even if we cannot offer a complete

236

description using a single framework (as in the case of smoothly covering a sphere with a

two-dimensional coordinate system) we can fit together our partial descriptions in a

consistent way to present a unproblematic picture of the whole.

We face a more fundamental worry when our limited descriptions are not of some

spatiotemporal parts of the whole, but rather of some limited range of the behavior that

the system is capable of. Effective field theories fall into this category, and, as our

discussion of the information loss paradox reveals, assessing the domains of validity of

such theories can be difficult and controversial. An understanding of the proposed

limitations of our physical descriptions is therefore necessary for an adequate account of

this particular debate in quantum gravity.

The importance of finding the proper boundaries of merely effective theories

extends well beyond the limited context of understanding this episode in the history of

physics, however. An adequate interpretation of either GR or QFT would also need to

offer an account of when and where these theories are no longer reliable. And if this is

true of our physical theories it is certainly also true in other sciences. Given that every

scientific theory we have is (at best) an effective theory – able to provide an adequate

description in certain situations, but ineffective in others – we should recognize that

erecting signposts in borderlands will be an essential part of laying claim to the

understanding of nature offered by our theories. That is, an essential part of our

interpretive efforts will be offering an account of the domain of applicability of our

theories. An interpretation should presumably tell us whether, how, and to what degree

the theory in question is applicable to the actual world. The question here is how a theory

38On this topic, see A. Bokulich (2001).

237

can be used to describe processes in our world – despite the fact that strictly speaking it is

a false theory at a fundamental level.

Complementarity is more than an account of theories that have a limited domain

of applicability, however. It is a more specific and somewhat subtle relationship between

two effective theoretical descriptions. Where one description becomes applicable the

complementary description becomes inapplicable. A full account of complementarity

requires us to specify three elements: the contexts in which each description is valid, the

limits at which each description breaks down, and the grounding of the incompatibility

between the contexts that secure the applicability of the complementary descriptions. We

need to do more than simply specify these elements: we should also justify our

characterization of these limits.

Let us consider each of these elements in turn as we consider Bohr’s interpretation

of QM in terms of complementarity. The applicability of classical concepts, such as

position, momentum, electric field values, etc. is to be secured by the correspondence

principle. Bohr’s account here is not as thorough as one might wish. The relationship

between classical and quantum mechanics remains problematic, as is evident, for

example, from recent work in quantum chaos.38 Bohr does not necessarily need a

complete description of the classical/quantum border, as he can rest content with the

slightly more modest claim that there are some situations that allow us to apply classical

mechanics unproblematically, and these are precisely the situations that we must arrange

if we wish to perform measurements. However, such an account places the burden of

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establishing the legitimacy of our effective descriptions on an account of measuring

arrangements; and, although Bohr has offered us a handful of examples, he has not

presented us with a general account of measurement that can clearly bear this burden.

Turning to the question of Bohr’s characterization of the breakdown of our

classical concepts, we again see that his position stands in need of support. In Section 6.2

we saw an argument for the claim that the foundation of Bohr’s belief in the

inapplicability of classical concepts was the existence of quantum phenomena for which

no classical model could be constructed. However, he also believed that where one is

able to apply classical theories – for example, in the context of a standard momentum

measurement – one is thereby able to apply the relevant classical concept; that is,

momentum measurements measure momentum. If one is willing to give up this premise,

as Bohm does, then one can retain the universal validity of a classical concept such as

position.

While this is not the place to enter into a debate over the strengths and weaknesses

of hidden-variable interpretations, it is important to point out that such a debate cannot be

avoided: Bohr has offered us no compelling reason for believing that the limits of

concepts such as position are given by the uncertainty relation. On the other hand, we

have also seen that Bohr has at least offered a consistent interpretation of the quantum

formalism, although the sacrifices that his account requires will lead many to question its

acceptability. For example, a scientific realist who is hoping for a complete theoretical

description of the real world is likely to find Bohr’s account unsatisfactory because it

39 Spontaneous localization models of quantum mechanics would be one example of sucha theory. Such models allow one to attribute properties to an object in a manner that isconceptually independent of the context that object happens to be in. See, for example,Ghirardi and Grassi (1996) and references therein.

239

establishes limitations to our descriptions without providing some further description that

will pick up where the classical one leaves off.

Such a realist need not be a determinist, for one might look for a stochastic

account that could still legitimately claim to be a (more) complete description of the

world.39 Einstein distinguishes between the issue of determinism and the issue of

describability in a note to Bohr dated April 4, 1949. The question that he sees at issue

between him and Bohr concerns “whether God is really playing dice and whether or not

we should hold on to the notion of a reality amenable to physical description” (quoted in

BCW 7: 281-282). Bohr’s lighthearted response very nicely captures the essence of the

explanatory power that he sees complementarity holding:

[I]t is not a question of whether or not we shall hold on to a notion of realityamenable to physical description, but rather of continuing along the road you haveshown us and of realizing the logical presuppositions for the description of therealities. . . . I may even say that nobody – not even the dear Lord himself – mayknow what an expression like playing dice means in this context. (BCW 7: 282)

The crucial point, according to Bohr, is that our descriptions in terms of quantities such as

positions and momenta assume the legitimacy of the classical physics. It might have been

the case that the application of these concepts would face no fundamental limitations, but

it has turned out that such universal applicability is ruled out by the nonclassical behavior

of quantum systems.

240

It is important to recognize that Bohr’s position primarily concerns epistemology

and not directly ontology. We should be wary of picturing objects possessing “fuzzy”

properties that are then somehow made concrete, suddenly snapped into focus, by a

measuring apparatus. Instead, we should think of descriptions in terms of position,

momentum, and so on, as being merely effective descriptions of objects, and the

effectiveness of these descriptions depends on the interaction of the object with an

apparatus of the appropriate sort. The attractiveness of such a position will largely

depend on what we expect our scientific theories to deliver.

In general, we should be willing to accept the fact that our descriptions, including

our physical descriptions, will only have a limited range of applicability. The shocking

aspect of Bohr’s claim is that there is no more fundamental description that will pick up

where the classical description gives out. Quantum theory gives us a statistical algorithm

that allows us to assess the probabilities of certain classically describable measurement

outcomes – but it does not give us a realistic description of the behavior of the object.

Such a description can be had if one rejects Bohr’s account of the limitations of classical

concepts – by adopting a position such as Bohm’s, for example. Such an account allows

one (in principle) to offer a complete description of the world, in the sense that we would

have a “notion of a reality amenable to physical description,” to repeat Einstein’s words.

On the other hand, if one rejects hidden variable theories such as Bohm’s (perhaps

because of the distasteful nonlocality that stands out when one accepts the validity of a

universal describability of position), and if one finds compelling Bohr’s account of the

legitimacy of attributing properties such as position and momentum when we have

40If one accepts a Bohrian account of attributing classical values, I believe that such anargument can be generated for any system that obeys an action principle. Building on thework of DeWitt (1962) one should be able to show that any context that allows theattribution of a classical value of the system (such a context would be required, forexample, for a measurement of that value) will render the context required for acomplementary context impossible. The argument would essentially be a generalizationof the Bohr and Rosenfeld (1933) argument that a measurement will introduce anuncontrollable reaction in any measurement of an incompatible observable. However,this remains a topic for future research.

241

secured the practical applicability of classical mechanics, then one may well be satisfied

with an account that requires us to forgo any description that cannot be offered in the

language of classical physics.

The third element of complementarity is the reciprocal incompatibility of the

descriptions in question. The importance of this aspect lies in the fact that if both

descriptions were applicable then there should be a classical description of the system,

which would be incompatible with the system’s actual behavior. Bohr’s account here

again rests on particular examples of our inability to produce arrangements that allow the

simultaneous application of complementary descriptions, and his further assertion that

this will always be the case. The essence of this incompatibility lies in the fact that there

will be an uncontrollable interaction between measurements of observables represented

by noncommuting operators. However, one would like a more general argument for the

inescapability of this conclusion, and this argument has yet to be produced.40 Further,

Bohr’s account again rests heavily on the claim that having an apparatus whose

functioning can legitimately be described (or predicted) using classical mechanics allows

us to attribute classical properties to a system.

242

We saw in Chapter 5 that most current arguments for BHC run afoul of this

criterion of the incompatibility of contexts allowing us to utilize the concepts that are

supposed to be complementary to one another. ’t Hooft’s account rests on an illegitimate

account of measurement collapse, and STU’s account makes a fallacious appeal to the

impossibility (or difficulty) of information transfer. KVV’s account postulates the

needed incompatibility but fails to offer a substantial justification for this postulate. Of

course, it may be that such a justification will require a full theory of quantum gravity,

just as we would not be able to establish the limits of classical mechanics if we did not

have the quantum formalism with its uncertainty relation. We have, at least, left open the

possibility that complementarity will provide an escape for unitary evolution from

Hawking’s paradox.

The considerations of Section 6.1 add some support to this possibility. The

arguable nonlocality of the quantum gravitational account offered by M-theory gives us

some reason to believe that a new form of noncommutativity may arise out of such a

theory, and if one accepts complementarity as an account of noncommuting operators in

standard QM, one should be able to extend this account to the quantum gravitational

context as well. One may speculate that because spacetimes can be considered to be

classical solutions to the Einstein field equations, a quantum theory of gravity should

admit superpositions of these classical solutions, i.e., superpositions of spacetimes.

Because local QFT is defined on a classical background spacetime, descriptions based on

such theories will be applicable only when we can legitimately ignore the quantum nature

of the spacetime – i.e., when superposition effects will be negligible. The question that

243

remains, however, is precisely when and where this will be the case. Arguments for black

hole complementarity give us reason to suspect that quantum gravitational effects may be

substantial enough to secure a unitary evolution for evaporating black holes and thus

resolve the information loss paradox.

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Figure A.1: Penrose conformal spacetime diagram.

APPENDIX A

CONFORMAL STRUCTURE AND PENROSE DIAGRAMS

Throughout this dissertation we represent spacetimes using Penrose conformal

diagrams. These diagrams represent the causal structure and the structure at infinity of

any spherically symmetric spacetime. As we shall see in slightly more detail below, the

null rays (light paths) of the spacetime are represented as lines of 45", regardless of the

curvature of the spacetime. This allows us to read the causal structure off the diagram

just as we do off standard Minkowski diagrams. The price that we pay for this

representation is the systematic distortion of all spatial and temporal intervals.

41 This presentation follows Hawking and Ellis (1973) pp. 120-123.

245

As an example, let us consider the Penrose diagram in Figure A.1, which can

equally well stand for Minkowski spacetime or a spacetime metric appropriate to a rigid

spherical object with a diameter larger than its Schwarzschild radius. In the latter case

the spacetime outside the matter (represented by the bold line) will be given by the

Schwarzschild metric. All timelike geodesic curves if extended into the infinite future

(respectively, past) will terminate at future (past) timelike infinity, signified by i+ (i!).

The diagram suppresses two spatial dimensions, which are of little interest anyway,

because the spacetimes under consideration are spherically symmetric. The remaining

spatial coordinate, r, reaches infinity at i0. Null rays that come in from infinity originate

on I! (past null infinity), and outgoing null rays reach infinity on I+ (future null infinity).

The justification for these characterizations can be seen as follows.41 Let us begin

with Minkowski spacetime:

ds2 = !dt2 + dr2 + r2(d2 2 + sin22 dN2).

In terms of our advanced and retarded null coordinates, defined as u = t + r and v = t !r,

we have

ds2 = !du2 + dv2 + ¼ (u ! v)2 (d2 2 + sin22 dN2).

Next we define a set of coordinates, p, q, that takes the infinities of the null coordinates u,

v to finite values:

tan p = u, tan q = v.

Here !½B < p < ½B; !½B < q < ½B; and p$q. The metric of Minkowski spacetime in

246

these coordinates will be

ds2 = sec2p sec2q (!dpdq + ¼ sin2(p!q) (d2 2 + sin22 dN2)). (A.1)

This form of our metric is of interest because it allows us to see that Minkowski

spacetime is conformal to a portion of an Einstein static spacetime. To say that two

metrics g:< and g):< are conformal to one another is to say that they are related as

g):< = S2 g:<

where S is any smooth positive function. This implies that the spacetimes defined by g:<

and g):< will have identical causal structures. An Einstein static universe is one in which

the cosmological constant is tuned to the critical value that prevents the universe from

expanding or contracting (i.e., it is static), the particular example of interest is given by

d)s2 = !(dtN)2 + (drN)2 + sin2rN(d2 2 + sin22 dN2).

Using the coordinate transformation tN = p + q, rN = p ! q, we can express this in a form

that is to be compared with the Minkowski metric given in (A.1):

d)s2 = !4dp dq + sin2(p!q) (d2 2 + sin22 dN2).

Thus the the Minkowski metric g:< is conformal to the metric g):< of a static Einstein

universe, as long as we restrict ourselves to a region of this universe given by

!B< tN+ rN<B, !B<tN!rN< B, rN $ 0.

Thus we find that the causal structure of Minkowski spacetime is equivalent to

that of a part of larger spacetime. If we suppress two dimensions of this Einstein static

spacetime, it can be represented as a cylinder in a three-dimensional spacetime. Our

entire Minkowski spacetime will then be conformal to the region !B<(tN±r’)<B of this

cylinder, as represented in Figure A.2. This construction allows us a natural

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Figure A.2: Einstein static spacetime.

representation of the conformal structure of infinity in Minkowski spacetime: we need

only consider the border of the Minkowski region.

The point i0 in Figure A.2 then represents the spatial border of Minkowski

spacetime. All maximally extended spacelike surfaces will intersect with this point.

Likewise, all timelike geodesics can be extended futureward to i+ and pastward to i!, and

all null geodesics will originate on I! (pronounced “scri-minus,” an abbreviation of

“script I minus”) and terminate on I+ (“scri-plus”). We can now peel the relevant portion

off the cylinder (figuratively speaking) and appeal to the spherical symmetry of the

spacetime to discard half of it, leaving us with the Penrose diagram of Figure A.1.

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APPENDIX B

MODELS OF BLACK HOLE COMPLEMENTARITY

In this appendix we present some of the details of the specific mathematical

models presented by the advocates of black hole complementarity. We begin with the

detailed toy model put forward by STU. This relies on a semi-classical model of black

holes developed by Callan et al. (1992), which is based on two-dimensional dilaton

gravity. The theory of dilaton gravity is a unified field theory that is the natural field

theoretic limit of string theory. The dilaton, M, is a scalar field that, in two dimensions,

encodes all the gravitational dynamics of the theory, because the two-dimensional

Einstein action, SE = IR d2x, is a topological invariant.

We begin with the action of classical two-dimensional dilaton gravity,

S0 = (1/2B) I d2x (!g)1/2 [ e-2M (R + 4(LM)2 + 482) ! Ei=1N (Lfi)

2],

where the fi are N massless noninteracting matter fields, and 82 is a cosmological

constant. Following Callan et al. (1992) we can consider a semiclassical extension of

this action, by incorporating the leading order quantum back reaction on the geometry of

the Hawking radiation produced in the matter fields. This first-order effect is given by

the Polyakov-Liouville term,

SL = !(N/96B) I d2x (-g(x))1/2 I d2xN (!g(xN))1/2 R(x) G(x;xN) R (xN),

249

where G is the Green’s function for the operator L2. The semiclassical theory then

follows from varying the action S = S0 + SL. Studies of this action have revealed that it

has solutions that describe evaporating black holes, although exact solutions have not

been found.

One can find exact solutions, however, by carefully modifying the dilaton

potential, or by adding specific terms to the semi-classical action. Russo, Susskind, and

Thorlacius (1992) find such exact solutions by inserting the following extra term by hand

into the effective action:

!(N/48)Id2x (!g)1/2 M R.

This then yields the toy model that STU suggest will be an adequate description of the

exterior of a two-dimensional black hole. The resulting effective action can be simplified

considerably if we work in lightcone coordinates, which here we shall follow STU in

writing as (y+, y!) rather than (U,V). Because any two-dimensional spacetime is

conformally flat, we can write the metric as

ds2 = !e2D dy+dy!.

We can then define two new fields, S and P, as

S = e!2M + 6M/2,

P = e!2M + 6(D !M/2).

This allows us to express our effective semiclassical action in the form,

Seff = B!1 Id2y [6!1(!M+P M!P + !M+S M!S) + e2(P!S)/6 + ½ Gi=1N M+fi M!fi],

where 6 = N/12.

250

Varying this action, we can find our equations of motion and constraint equations,

which are given by

M+ M!P = M+ M!S = e2(P!S)/6,

6!1 ((M+S)2 ! (M+P)2) M+2P + T+ + ! 6t+ = 0,

6!1 ((M!S)2 ! (M!P)2) M!2P + T!! ! 6t! = 0.

Here t+ and t! are functions of integration that are fixed by the boundary conditions. (See

Callan et al. 1992, p. R1007 for a discussion of the properties of these functions.)

Solutions to these equations of motion describe the geometry of our two-dimensional

spacetime, including the semiclassical back reaction of the Hawking radiation.

We thus have a highly idealized model that allows us to investigate how black

holes evolve when giving off radiation. As expected, we find that during this process the

area of the black hole shrinks. We can also follow the evolution of the curvature

singularity and apparent horizon through the spacetime and determine the position of the

event horizon. A troublesome feature of these evolutions is the fact that the singularity

does not remain behind the apparent horizon, but becomes a naked singularity exposed to

external observers. Russo, Susskind, and Thorlacius argued in their (1992, 1993) that the

curvature singularity indicates the border of applicability of the semiclassical model. To

use it as an effective model one would have to impose boundary conditions along this

curve, requiring that the curvature remain finite along this boundary. This model had

some unpleasant features, however, including a difficulty in allowing for energy

conservation. Russo, Susskind, and Thorlacius were forced to postulate that the

251

evaporation of a black hole would end with a “thunderpop” of negative energy that would

balance the conservation books.

Many of the problematic features of this model are eliminated if one imposes an

appropriate set of boundary conditions at the stretched horizon of the black hole rather

than at the singularity. STU argued that the semiclassical description of the evolution of

the two-dimensional spacetime was physically reasonable – as long as one only trusted

the theory outside of the stretched horizon, and imposed appropriate boundary conditions

at that surface. This is then the semiclassical theory that they propose in their second

postulate of Black Hole Complementarity.

Our second semiclassical calculation offered in support of black hole

complementarity is KVV’s first order calculation of the gravitational interaction between

ingoing and outgoing modes of a free (apart from leading order gravitational effects)

field. We consider a Schwarzschild spacetime as depicted in Figure 3.3 with a Cauchy

surface I+chS, which consists of future null infinity and the event horizon of the black

hole. Using the null and angular coordinates (u,v,S), KVV represent the gravitational

interaction by the following time-ordered action:

This will allow us to represent the adjusted final state as |R,final = exp(i Sint)|R,Hawking. To

get a sense of the size of these leading order corrections we need to calculate the

magnitude of Tuu and Tvv for the Hawking state.

252

Following KVV we designate out modes defined at I+ as bT and horizon modes

defined near hS as cT, and define a coherent basis for our final Hilbert space as

|n, = exp [I dT(nT bT† + n!T cT

†)] |0,bq|0,c.

We can then express the components of the Hawking state as

+RHawking | n, = N exp [iIdT e!4BMT nT n!T],

where N is a normalization constant. The incoming stress energy component of the

Hawking state can be written as

where

We can see that Tvv is generally singular at v = 0, which is the null trajectory we find by

extending the event horizon back to I+ (see Figure 3.3). Likewise, KVV report, Tuu –

which is related to Tvv by reflection off the origin, r=0 – is generally singular as we

approach the horizon hS, “as soon as the nT differ by a small amount from their exact

expectation value” (p. 7063). Our effective semiclassical theory will therefore require us

to cut off our gravitational interaction some reasonable distance above the horizon and

above v=0.

As discussed in Section 5.5, it is the necessity of using incompatible cutoffs to

secure the applicability of our semiclassical models that drives KVV’s claim that the

descriptions offered in terms of these models will be complementary. We can trust our

253

semiclassical calculation on hS if and only if we impose a short-distance cutoff near the

horizon, but this will not be compatible with a legitimate description of the field on I+. If

we instead wish to have an accurate description of the outgoing radiation at I+, then we

will have to retain high-energy modes near the horizon, which (according to KVV) will

rule out the possibility of an accurate semiclassical account of the region near the horizon.

254

BIBLIOGRAPHY

Aharonov, Y. and Bohm, D. (1961). “Time in the Quantum Theory and the UncertaintyRelation for Time and Energy.” Physical Review 122: 1649-58.

Ballentine, L. (1990). Quantum Mechanics. Englewood Cliffs, NJ: Prentice Hall.

Banks, T. (1995). “Lectures on Black Holes and Information Loss.” Nuclear Physics B(Proceedings Supplement) 41: 21-65.

Banks, T., Susskind, L. and Peskin, M. E. (1995). “Difficulties for the Evolution of PureStates into Mixed States.” Nuclear Physics B 244: 125-134.

Bardeen, J. M., Carter, B., and Hawking, S. W. (1973). “The Four Laws of Black HoleMechanics.” Communications in Mathematical Physics 31:161-170.

Baym, G. (1969). Lectures on Quantum Mechanics. New York: Benjamin.

Bekenstein, J. D. (1973). “Black Holes and Entropy.” Physical Review D 7: 2333-2346.

__________(1981). “Universal Upper Bound on the Entropy-to-Energy Ratio forBounded Systems.” Physical Review D 23: 287-298.

Bell, J. S. (1987). Speakable and Unspeakable in Quantum Mechanics. New York:Cambridge UP.

Beller, M. (1992). “The Birth of Bohr's Complementarity: the Context and theDialogues.” Studies in the History and Philosophy of Science 23: 147-180.

_______ (1999). Quantum Dialogue: The Making of a Revolution. Chicago: U ofChicago P.

Beller, M. and Fine, A. (1994). “Bohr’s Response to EPR.” In Faye and Folse (1994)1-31.

255

Belot, G., Earman, J., and Ruetsche, L. (1999). “The Hawking Information Loss Paradox:The Anatomy of a Controversy.” British Journal for the Philosophy of Science 50:189-229.

Birrell, N. D. and Davies, P. C. W. (1982). Quantum Fields in Curved Space. New York:Cambridge UP.

Bohm, D. (1952). “A Suggested Interpretation of the Quantum Theory in Terms of‘Hidden’ Variables, I and II.” Physical Review 85, 166-193.

Bohr, N. (1928). “The Quantum Postulate and the Recent Development of AtomicTheory.” Nature 121: 580-590. Reprinted in Bohr (1985) 148-158.

________(1929a). “The Atomic Theory and the Fundamental Principles Underlying theDescription of Nature.” Address to the 18th Meeting of Scandinavian Scientists. In Bohr (1934) 102-119. Reprinted in Bohr (1985) 236-253.

________(1929b). “The Quantum of Action and the Description of Nature.” In Bohr(1934) 92-101. Reprinted in Bohr (1985) 208-217. Translation of“Wirkungsquantum und Naturbeschreibung.” Naturwissenschaften 17: 483-486.

________(1931). “Maxwell and Modern Theoretical Physics.” Nature (Supplement) 128:691-692.

________(1932). “Faraday Lecture: Chemistry and the Quantum Theory of AtomicConstitution.” Journal of the Chemical Society, 1932, pp. 349-384. Reprinted inBohr (1985) 373-408.

________(1934). “Introductory Survey” (with “Addendum of 1931”), of Atomic Theoryand the Description of Nature. New York: Cambridge UP. 1-24. Reprinted inBohr (1985) 279-302.

________(1935). “Can Quantum-Mechanical Description of Physical Reality BeConsidered Complete?” Physical Review 48, 696-702. Reprinted in Bohr (1996)291-298.

________(1939). “The Causality Problem in Atomic Physics.” In New Theories inPhysics. Paris: International Institute of Intellectual Co-operation. 11-45. Reprinted in Bohr (1996) 303-322.

________(1948). “On the Notions of Causality and Complementarity.” Dialectica 2: 312-319. Reprinted in Bohr (1996) 325-338.

256

________(1949). “Discussion with Einstein on Epistemological Problems in AtomicPhysics.” In P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist. LaSalle,Illinois: Open Court. 201-241. Reprinted in Bohr (1996) 339-381.

________(1958). “On Atoms and Human Knowledge.” Dædalus: Proceedings of TheAmerican Academy of Arts and Sciences 87: 164-175. Reprinted in Bohr (1996)412-423.

________(1961). “The Rutherford Memorial Lecture 1958: Reminiscences of theFounder of Nuclear Science and of some Developments based on his Work.”Proceedings of the Physical Society 78: 1083-1115. Reprinted in Bohr (1999)383-415.

________(1985). Niels Bohr: Collected Works, Vol. 6. Foundations of Quantum Physics I(1926-1932). Ed. Jørgen Kalckar. New York: North-Holland.

________(1996). Niels Bohr: Collected Works, Vol. 7. Foundations of Quantum PhysicsII (1933-1958). Ed. Jørgen Kalckar. New York: North-Holland.

________(1999). Niels Bohr: Collected Works, Vol. 10. Complementarity BeyondPhysics (1928-1962). Ed. Finn Aaserud. New York: North-Holland.

Bohr, N., Kramers, H. A., and Slater, J. C. (1924). “The Quantum Theory of Radiation.”Philosophical Magazine 47: 785-802.

Bohr, N. and Rosenfeld, L. (1933). “On the Question of the Measurability ofElectromagnetic Field Quantities.” Trans. A. Peterson. In Cohen and Stachel(1979): 357-400. Reprinted in Bohr (1996) 123-166. Trans. of "Zur Frage derMessbarkeit der Elektromagnetischen Feldgrössen." Det Kongelige DanskeVidenskabernes Selkab, Mathmatisk-fysiske Meddelelser 12: 1-65. Reprinted inBohr (1996) 57-121.

_____________(1950). “Field and Charge Measurements in Quantum Electrodynamics.”Physical Review 78: 794-898. Reprinted in Bohr (1996) 211-216.

Bokulich, A. (2001). “Philosophical Perspectives on Quantum Chaos: Models andInterpretations." Ph.D. Dissertation, University of Notre Dame.

Bokulich, P. (2001). “Black Hole Remnants and Classical vs. Quantum Gravity.”Philosophy of Science 68: S407-S423.

Bridgman, P. W. (1927). The Logic of Modern Physics. New York: MacMillan.

257

Brown, H. and Harré, R., eds. (1988). Philosophical Foundations of Quantum FieldTheory. Oxford: Clarendon.

Brown, H. and Redhead, M. (1981). “A Critique of the Disturbance Theory ofIndeterminacy in Quantum Mechanics.” Foundations of Physics 11: 1-20.

Busch, P. and Lahti, P. J. (1984). “On Various Joint Measurements of Position andMomentum Observables in Quantum Theory.” Physical Review D 29: 1634-1646.

Byron, F. W. and Fuller, R. W. (1992 [1969,1970]). Mathematics of Classical andQuantum Physics. Two volumes bound as one. New York: Dover.

Callan, C., Giddings, S. B., Harvey, J., Strominger, A. (1992). “Evanescent Black Holes.” Physical Review D 45: R1005-R1009.

Callender, C. and Huggett, N. (2001). Physics Meets Philosophy at the Planck Scale. New York: Cambridge UP.

Caves, C., Thorne, K., Drever, R., Sandberg,, V., and Zimmermann, M. (1980). “On theMeasurement of a Weak Classical Force.” Reviews of Modern Physics 52:341-392.

Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes. New York:Oxford UP.

Christensen, S. M., ed. (1984). Quantum Theory of Gravity: Essays in Honor of the 60thBirthday of Bryce S DeWitt. Bristol: A. Hilger.

Clifton, R. (2000). “Complementarity between Position and Momentum as aConsequence of Kochen-Specker Arguments.” Physics Letters A 271:1-7.

Clifton, R. and Halvorson, H. (2002). “Reconsidering Bohr's Reply to EPR.” In J.Butterfield and T. Placek, eds. Non-locality and Modality. Dordrecht: Kluwer.

Cohen, R. S. and Stachel, J. J., eds. (1979). Selected Papers of Léon Rosenfeld. Boston: Reidel.

Cushing, J. T. (1988). “Foundational Problems in and Methodological Lessons fromQuantum Field Theory.” In Brown and Harré (1998) 25-39.

_________(1994a). “A Bohmian Response to Bohr's Complementarity.” In Faye andFolse (1994) 57-76.

258

________(1994b). Quantum Mechanics: Historical Contingency and the CopenhagenHegemony. Chicago: U. of Chicago P.

Cushing, J.T., Fine, A., and Goldstein, S. (1996). Bohmian Mechanics and QuantumTheory: An Appraisal. Dordrecht: Kluwer.

Cushing, J. T. and McMullin, E., eds. (1989). Philosophical Consequences of QuantumTheory: Reflections on Bell's Theorem. Notre Dame: U. Notre Dame.

Darrigol, O. (1991). “Cohérence et Complétude de la Mécanique Quantique: l’exemplede Bohr-Rosenfeld.” Revue d'Histoire des Sciences 44(2):137-179.

Davies, P. C. W. (1984). “Particles Do Not Exist.” In Christensen (1984) 66-77.

de Haro, S. (2001). “Quantum Gravity and the Holographic Principle.” Ph.D.Dissertation, Universiteit Utrecht.

DeWitt, B. S. (1962). “Definition of Commutators via the Uncertainty Principle.” Journal of Mathematical Physics 3: 619-624.

Dickson, M. (2003). “Quantum Reference Frames in the Context of EPR.” Forthcomingin Proceedings of the 2002 Philosophy of Science Association Meeting.

Earman, J. (1986). A Primer on Determinism. Boston: Reidel.

________(1995). Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalitiesin Relativistic Spacetimes. New York: Oxford UP.

Einstein, A., Podolsky, B., and Rosen, N. (1935). “Can Quantum-Mechanical Descriptionof Physical Reality Be Considered Complete?” Physical Review 47: 777-80. Reprinted in Wheeler and Zurek (1983) 138-141.

Faye, J. and Folse, H. J., eds. (1994). Niels Bohr and Contemporary Physics. Dordrecht:Kluwer.

Fine, A. (1986). The Shaky Game: Einstein, Realism, and the Quantum Theory. Chicago:U of Chicago P.

Fré, P., Gorini, V., Magli, G., and Moschella, U. (1999). Classical and Quantum BlackHoles. Philadelphia: Institute of Physics.

Frisch, M. (2003). Philosophy and Classical Electrodynamics (forthcoming).

259

Fulling, S. A. (1989). Aspects of Quantum Field Theory in Curved Space-Time. NewYork: Cambridge UP.

Ghirardi, G. and Grassi, R. (1996). “Bohm’s Theory versus Dynamical Reduction.” InCushing et al. (1996) 353-377.

Giddings, S. B. (1992). “Black Holes and Massive Remnants.” Physical Review D 46:1347-52.

__________(1995a). “Quantum Mechanics of Black Holes.” In E. Gava, A. Masiero, K.S. Narain, S. Randjbar-Daemi, and Q. Shafi, eds. 1994 Summer School in HighEnergy Physics and Cosmology. ICTP Series in Theoretical Physics-Vol. 11.Singapore: World Scientific. 530-74.

__________(1995b). “Why Aren’t Black Holes Infinitely Produced?” Physical Review D51: 6860-6869.

Gillies, D. A. (1972). “Operationalism.” Synthese 25: 1-24.

Haag, R. (1992). Local Quantum Physics: Fields, Particles, Algebras. New York: Springer-Verlag.

Halvorson, H. (2000). “The Einstein-Podolsky-Rosen State Maximally Violates Bell’sInequalities.” Letters in Mathematical Physics 53: 321-329.

____________(2003). “Complementarity of Representations in Quantum Mechanics.” Forthcoming in Studies in History and Philosophy of Modern Physics.

Hawking, S. W. (1974). “Black Hole Explosions?” Nature 248: 30-31.

____________(1975). “Particle Creation by Black Holes.” Communications inMathematical Physics 43: 199-220.

____________ (1976). “The Breakdown of Predictability in Gravitational Collapse.” Physical Review D 14: 2460-2473.

____________ (1982). “The Unpredictability of Quantum Gravity.” Communications inMathematical Physics 87: 395-415.

Hawking, S. W. and Ellis, G. F. R. (1973). The Large Scale Structure of Spacetime. NewYork: Cambridge UP.

Heitler, W. (1944). The Quantum Theory of Radiation. New York: Oxford UP.

260

Henley, E. and Thirring, W. (1962). Elementary Quantum Field Theory. New York: McGraw-Hill.

Holland, P. R. (1993). The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. New York: Cambridge UP.

Howard, D. (1979). “Complementarity and Ontology: Niels Bohr and the Problem ofScientific Realism in Quantum Physics.” Ph.D. Dissertation, Boston University.

__________(1989). “Holism, Separability, and the Metaphysical Implications of the BellExperiments.” In Cushing and McMullin (1989) 224-253.

__________(1990). “‘Nicht Sein Kann Was Nicht Sein Darf,’ or the Prehistory of EPR,1909-1935: Einstein’s Early Worries about the Quantum Mechanics of CompositeSystems.” In Miller (1990) 61-111.

__________(1994). “What Makes a Classical Concept Classical?: Toward aReconstruction of Niels Bohr's Philosophy of Physics.” In Faye and Folse (1994)201-229.

__________(2003). “Who Invented the Copenhagen Interpretation? A Study inMythology.” Forthcoming in Proceedings of the 2002 Philosophy of ScienceAssociation Meeting.

Huggett, N. (1996). “Exposing the Machinery of Infinite Renormalization.” Proceedingsof the 1996 Philosophy of Science Association Meeting, Vol. 1. 159-167.

Kamefuchi, S., et al., eds. (1984). Foundations of Quantum Mechanics in the Light ofNew Technology. Tokyo: Physical Society of Japan.

Kiem, Y., Verlinde, H., and Verlinde, E. (1995). “Black Hole Horizons andComplementarity.” Physical Review D 52: 7053-7065.

Landau, L. and Peierls, R. (1931). “Erweiterung des Unbestimmtheitsprinzips für dieRelativistische Quantentheorie.” Zeitschrift für Physik 69: 56-69.

Laplace, P. (1796). Exposition du System du Monde. Paris: Cercle-Social.

Lowe, D., Polchinski, J., Susskind, L., Thorlacius, L., and Uglum, J. (1995). “Black HoleComplementarity versus Locality.” Physical Review D 52, 6997-7010.

Lowe, D. and Thorlacius, L. (1999). “AdS/CFT and the Information Paradox,” PhysicalReview D 60: 104012-1 to 104012-7.

261

Michell, J. (1784). “On the Means of discovering the Distance, Magnitude, &c. of theFixed Stars, in consequence of the Diminution of the velocity of their Light, incase such a Diminution should be found to take place in any of them, and suchData should be procurred from Observations, as would be farther necessary forthat Purpose.” Philosophical Transactions 74: 35-57.

Miller, A. I., ed. (1990). Sixty-Two Years of Uncertainty: Historical, Philosophical andPhysical Inquiries into to the Foundations of Quantum Mechanics. New York: Plenum.

Page, D. (1980). “Is Black-Hole Evaporation Predictable?” Physical Review Letters 44: 301-304.

________(1994). “Black Hole Information.” In R. B. Mann and R. G. McLenaghan, eds.Proceedings of the 5th Canadian Conference on General Relativity andRelativistic Astrophysics. London: World Scientific. 1-41.

Peskin, M. E. and Schroeder, D. V. (1995). An Introduction to Quantum Field Theory.New York: Addison-Wesley

Polchinski, J. (1998a). String Theory. Volume I. New York: Cambridge UP.

___________(1998b). String Theory. Volume II. New York: Cambridge UP.

Quine, W. V. (1951). “Two Dogmas of Empiricism.” Philosophical Review 60: 20-43. Reprinted in W. V. Quine, From a Logical Point of View. Cambridge, MA:Harvard UP (1953) 20-46.

Rosenfeld, L. (1953). “Strife about Complementarity.” Science Progress 41: 393-410. Reprinted in Cohen and Stachel (1979) 465-483.

___________(1961). “Foundations of Quantum Theory and Complementarity.” Nature190: 384-388.

___________(1963).“On Quantization of Fields.” Nuclear Physics 40: 353-356. Reprinted in Cohen and Stachel (1979) 442-445.

Russo, J., Susskind, L., and Thorlacius, L. (1992). “Black Hole Evaporation in 1+1Dimensions.” Physical Review D 46: 3444-3449.

Ryder, L. H. (1985). Quantum Field Theory. New York: Cambridge UP.

262

Saunders, S. and Brown, H. R., eds. (1991). The Philosophy of Vacuum. New York:Oxford UP.

Shimony, A. (1984). “Controllable and Uncontrollable Non-locality.” In Kamefuchi et al. (1984) 225-230.

Sklar, L. (1993). Physics and Chance: Philosophical Issues in the Foundations ofStatistical Mechanics. New York: Cambridge UP.

Stephans, C. R., ’t Hooft, G., and Whiting, B. F. (1994). “Black Hole EvaporationWithout Information Loss.” Classical and Quantum Gravity 11: 621-647.

Strominger, A. and Vafa, C. (1996). “Microscopic Origin of the Bekenstein-HawkingEntropy.” Physics Letters B 379: 99-104.

Susskind, L. (1970). “Structure of Hadrons Implied by Duality” Physical Review D 1: 1182-1186.

_________(1993). “String Theory and the Principle of Black Hole Complementarity.” Physical Review Letters 71: 2367-2368.

_________(1994). “Strings, Black Holes, and Lorentz Contraction.” Physical Review D49: 6606-6611.

_________(1995). “The World as a Hologram.” Journal for Mathematical Physics 36:6377-6396.

_________(1997). “Black Holes and the Information Paradox.” Scientific American,272, 4, April: 52-7.

Susskind, L. and Thorlacius, L. (1994). “Gedanken Experiments Involving Black Holes.”Physical Review D 49: 966-974.

Susskind, L., Thorlacius, L., and Uglum, J. (1993). “The Stretched Horizon and BlackHole Complementarity.” Physical Review D 48: 3743-3761.

Susskind, L., and Uglum, J. (1996). “String Physics and Black Holes.” Nuclear Physics B(Proceedings Supplement) 45: 115-134.

Tanona, S. (2003). “Idealization and Formalism in Bohr’s Approach to QuantumTheory.” Forthcoming in Proceedings of the 2002 Philosophy of ScienceAssociation Meeting.

263

Teller, P. (1995). An Interpretative Introduction to Quantum Field Theory. Princeton:Princeton UP.

’t Hooft, G. (1985). “On the Quantum Structure of a Black Hole.” Nuclear Physics B256: 727-745.

__________(1987). “Strings from Gravity.” Physica Scripta T 15 143-150.

__________(1988). “Black Holes and the Foundations of Quantum Mechanics.” In H.Feshbach, T. Matsui, and A. Oleson, eds. Niels Bohr: Physics and the World. New York: Harwood. 171-182.

__________(1990). “The Black Hole Interpretation of String Theory.” Nuclear Physics B335:138-154.

__________(1996). “The Scattering Matrix Approach for the Quantum Black Hole:an Overview.” International Journal of Modern Physics A 11: 4623-4688.

__________(1999). “Quantum Gravity as a Dissipative Deterministic System.” Classicaland Quantum Gravity 16: 3263-3279.

__________(2001). “Determinism and Dissipation in Quantum Gravity.” In A. Zichichi,ed. Basics and Highlights in Fundamental Physics, The Subnuclear Series–Vol.37, London: World Scientific. 397-430.

__________(2002). “How Does God Play Dice? (Pre-)Determinism at the Planck Scale.” In R.A. Bertlmann and A. Zeilinger, eds. Quantum [Un]speakables, From Bell toQuantum Information. New York: Springer-Verlag. 307-316.

Thorlacius, L. (1995). “Black Hole Evolution.” Nuclear Physics B (ProceedingsSupplement) 41: 245-275.

Thorne, K., Price, R., and Macdonald, D. (1986). Black Holes: The Membrane Paradigm.New Haven: Yale UP.

Uffink, J. B. M. and Hilgevoord, J. (1985). “Uncertainty Principle and UncertaintyRelations.” Foundations of Physics 15: 925-944.

Unruh, W. (1976). "Notes on Black Hole Evaporation." Physical Review D 14: 870-892.

Unruh, W. and Wald, R. M. (1982). “Acceleration Radiation and the Generalized SecondLaw of Thermodynamics.” Physical Review D 25: 942-958.

264

___________(1984). “What Happens When an Accelerating Observer Detects a RindlerParticle.” Physical Review D 29: 1047-1056.

___________(1995). “Evolution Laws Taking Pure States to Mixed States in QuantumField Theory.” Physical Review D 52: 2176-2182.

Wald, R. M. (1984). General Relativity. Chicago: U of Chicago P.

___________(1994). Quantum Field Theory in Curved Spacetimes and Black HoleThermodynamics. Chicago: U of Chicago P.

___________(2001). “The Thermodynamics of Black Holes.” Living Reviews inRelativity 4,6: 1-44. [Online Article]: cited on 10/20/2001,http://www.livingreviews.org/Articles/Volume4/2001-6wald/.

Wessels, L. (1989). “The Way the World Isn’t: What the Bell Theorems Force us to GiveUp.” In Cushing and McMullin (1989) 80-96.

Wheeler, J. A. and Zurek, W., eds. (1983). Quantum Theory and Measurement. Princeton: Princeton UP.