a local spectral condition for axiomatic quantum fields on the de sitter surface

8/6/2019 A Local Spectral Condition for Axiomatic Quantum Fields on the de Sitter Surface

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A Local Spectral Condition for Axiomatic Quantum Fields

on the de Sitter Surface

by

Justin Manning

(Under the direction of Dr. Robert Varley)

Abstract

The Wightman axioms provide an intuitive and mathematically rigorous approach to quan-

tum field construction on flat Minkowski space, and much work has been done to extend these

axioms to curved space-time manifolds. A key axiom in the Wightman framework states that

the generator of time translation must be a positive semi-definite operator, which guarantees

non-negative energy for a quantum field. One of the major difficulties in extending these

axioms to curved manifolds arises when the manifold in question has no global time-like

isometry, and hence no global notion of time and energy. I propose a set of Wightman-like

axioms for the de Sitter surface which includes a solution to this problem by introducing

a local non-negative spectrum condition which gives a local meaning to energy, in essence

a local Hamiltonian, and guarantees it to be a non-negative operator. I then construct a

mathematically rigorous free massless scalar field on two-dimensional de Sitter space and

prove that it satisfies these axioms.

Index words: quantum field theory, curved space time, diagonally dominant, localspectrum, Wightman axioms, infinite dimensional matrix


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A Local Spectral Condition for

Axiomatic Quantum Fields on the de Sitter Surface

by

Justin Manning

B.S., University of Georgia, 2002

A Dissertation Submitted to the Graduate Faculty

of The University of Georgia in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Athens, Georgia

2011


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c2011Justin Manning

All Rights Reserved


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A Local Spectral Condition for Axiomatic Quantum Fields

on the de Sitter Surface

by

Justin Manning

August 6, 2011

Approved:

Major Professor: Robert Varley

Committee: Ed Azoff Clint McCroryMitch RothsteinMichael Usher

Electronic Version Approved:

Maureen GrassoDean of the Graduate SchoolThe University of GeorgiaAugust 2011


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Acknowledgments

Foremost, I would like to thank my advisor, Dr. Robert Varley for uncountable hours of work

and support. This thesis would not be possible were it not for his patience and guidance.

I would also like to thank my committee, Dr. Ed Azoff, Dr. Clint McCrory, Dr. Mitch

Rothstein, and Dr. Michael Usher for their time and input. Each has provided valuable

insight and feedback during the development and implementation of the ideas of this thesis

through conversations, seminar presentations, and corrections.

Last, I would like to thank my family and friends for their continued support and encour-

agement. In particular, Justine Samawicz has been my greatest supporter and encourager

throughout these studies.

Mathematics is not a careful march down a well-cleared highway, but a journey into a

strange wilderness, where the explorers often get lost. Rigour should be a signal to the

historian that the maps have been made, and the real explorers have gone elsewhere.

-W.S. Anglin

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Contents

1 Introduction 1

1.1 Minkowski Space and the Wightman Axioms . . . . . . . . . . . . . . . . . . 1

1.2 Segal Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Example: Free Massless Scalar Quantum Field on 2-D Minkowski Space . . . 8

2 Axiomatic Quantum Fields on the de Sitter Surface 14

2.1 The de Sitter Surface and its Axioms . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Example: Free Massless Scalar Quantum Field on the de Sitter Surface . . . 20

2.3 Axioms CW1-CW3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Diagonal Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 CW5, the Local Spectral Condition . . . . . . . . . . . . . . . . . . . . . . . 38

3 Connections and Speculations 43

3.1 Connections to Other Work in the Field . . . . . . . . . . . . . . . . . . . . 43

3.2 Speculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 49

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Chapter 1

Introduction

1.1 Minkowski Space and the Wightman Axioms

The Wightman axioms [11, 21] provide an intuitive and elegant mathematical definition of a

quantum field on Minkowski space. Each axiom corresponds to a natural physical idea and

gives a mathematical requirement for each that a quantum field must satisfy. Throughout

this thesis, the quantum fields under discussion are assumed to be scalar, in that they

have no spin. Scalar fields successfully demonstrate the essential structure of the Wightman

quantum field theory without the added complexity of spin. Before diving into the Wightman

axioms, let us review the underlying structure of Minkowski space on which it is defined.

N-dimensional Minkowski space is the space-time manifold that provides a mathe-

matical model of special relativity. It is defined as M = RN with the metric

v, w = v0w0 v1w1 v2w2 vN1wN1

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or in matrix form

=

1 0 0 0 1 0 0 0

1

.

The causal influence of a vector v is called the light cone ofv, denoted Cv, given by the

set

Cv = {w M|v w, v w 0}.

Two vectors v and w are called space-like separated if w / Cv, or equivalently v /

Cw, time-like separated if w Int(Cv) (or equivalently v Int(Cw)), and light-likeseparated if v Cw (or equivalently w Cv). These terms characterize the causalrelationship between v and w. For example, if v and w are space-like separated, then one

cannot be reached from the other without surpassing the speed of light, but if they are

time-like separated, one can be reached from the other by a speed less than the speed of

light.

The Lorentz group

L= O(1, N

1) is the group of linear isometries of M. It can be

characterized as

L = {G GL(RN)|GT G = }.

This group has four connected components. Write a matrix in L as

T =

a00 a01 a02 a10 a11 a12 a20 a21 a22

.

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The four components ofL are characterized by

L0 = {T L | a00 > 0,det(T) > 0}

L1 = {T L | a00 < 0,det(T) > 0}L2 = {T L | a00 > 0,det(T) < 0}

L3 = {T L | a00 < 0,det(T) < 0}.

For quantum fields on Minkowski space, we will be interested in L0, the identity compo-nent ofL. Viewing time as the first coordinate ofM, this subgroup consists of the isometriesofM that preserve both orientation on M and time direction. These are the transformations

that physics requires to preserve any physical measurement.

These, however, are not the only transformations that we need. Translations in M should

also preserve any physical measurements. Well define the Poincare group as

P= RN L0

where we use the common abuse of notation and write RN for the translations on RN.

Pwill be the group with which we are concerned while building a quantum field. We will writean element ofPas (a, ) where a RN and L0.

As in standard treatments of quantum mechanics, the states of a quantum field will

be modeled by a separable Hilbert space, and the inner product will be used to model

measurements. Thus for a quantum field, as in standard quantum mechanics, we will require

a unitary representation ofPon the Hilbert space of states.

The purpose of a quantum field is to model changing numbers of particles, so for each

point of M, we would ideally like an operator on the Hilbert space of states that models

creation and annihilation of particles at that point. Such a map on points of M is unfortu-

nately too singular, so instead we use a map on compactly supported functions of M. This

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corresponds to physics experiments, however, as measurements cannot take place at a point,

but only on an area ofM.

To this end, we define an operator valued distribution on M as a linear map

: C0 (M) End(D)

where D H is a dense linear subspace of a separable Hilbert space H, and where the map

f 1, (f)2

is a continuous linear functional for each 1, 2 D, using the Schwartz topology on C0 (M).We require a dense subspace because in most examples, (f) is unbounded.

In their famous work, Garding and Wightman [11] formulated a set of physically mean-

ingful axioms that these two ingredients of a quantum field should satisfy. A quantum field

in the Wightman axioms is a quadruple {, U, F, F0}, where F is a separable Hilbertspace, F0 a dense linear subspace, is an operator valued distribution on M with values inEnd(

F0), and U is a unitary representation of

Pon

F, where by convention, when we refer

to a unitary representation of a Lie group, we imply that it is strongly continuous. This

quadruple in addition must satisfy the following axioms.

W1: (Symmetry) 1, (f)2 = (f)1, 2 for all f C0 (M) and 1, 2 D.

W2: (Vacuum Cyclicity) There is a U-invariant vector F0 with |||| = 1 suchthat the linear span of vectors of the form (f1) (fn) for fj C0 (M), n N is

dense in F0

W3: (Equivariance) U(g)(f)U(g)1 = (f g1) for all f C0 (M) and g P

W4: (Causality Commutativity) [(f1), (f2)] = 0 ifsupp(f1) and supp(f2) are space-

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like separated

W5: (Spectral Condition) The infinitesimal generator H of U((t, 0, ), I), the timetranslation subgroup, has non-negative spectrum.

A quick note on convention: we will regard N as the positive integers not including 0.

Also, the symbol I will represent the identity map in its context.

These axioms correspond to general physical considerations. W1 models the idea that we

can generate particles from the vaccum by exchange of energy and matter, and create any

state of a finite number of particles. W2 guarantees that transformations of the space-time

M correspond to transformations of measurements of states in

H. W3 gives some control

of the spectrum of operators given by . In particular, if f is real-valued, then (f) has

real spectrum and can be used as an observable operator. W4 is the causality condition

that places of space-time that cannot reach each other by the speed of light cannot affect

measurements in their regions. W5 is the condition that the energy of a configuration of

states should be non-negative.

Note that axiom W5 is stated in a coordinate dependent way. This matches the usual

statements of the spectral axiom, however, because U((t, 0, ), I) is conjugate to any trans-lation in the forward light cone by elements of L0. Thus the generator of any translation inthe forward light cone will have a non-negative spectrum.

1.2 Segal Quantization

One helpful tool for constructing a Wightman quantum field is Segal quantization [20].

This gives a process by which to turn a one-particle quantum system into a many-particle

system. Let H be a separable Hilbert space. Well think of H as the space representing thestates of a one particle quantum system.

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Consider the symmetric projection Sn : Hn Hn given by

Sn(h1 hn) = 1n!

Sym(n)

h(1) h(n).

To construct the state space for n of these particles, we use the Hilbert space symmetric

tensor product SnHn. The tensor is symmetrized because identical particles are indistin-guishable. The state space for a quantum field must contain any number of particles, so we

define the following.

Let F0 = C (n=1SnHn), extending the inner product of H in the natural way overtensors and sums. F0 is called the finite particle space, containing finite linear combina-tions of any number of particles. The Hilbert space closure F= F0 is called the symmetricFock space on H. The vector = (1, 0, 0, ) is called the vacuum, representing a statewith no particles.

Now we perform a similar construction for self-adjoint operators on H. If A is a self-adjoint operator on H, we define (A) on the n particle space SnHn by

(A) = A I

I+I A

I

I+ +

I

I A

and extend to F0 by linearity. The map is called the second quantization of A.If U is a unitary operator on H, then we define E(U) on SnHn by

E(U) = nk=1U

and extend by linearity to F. Cleary, then, E(U) is unitary on F.When constructing a quantum field by Segal quantization, Fwill be the Hilbert space of

states of the system, and F0 will be the dense subspace referenced in the Wightman axioms.Also, we must model creation and annihilation of particles to represent changes in the field.

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This is accomplished through creation and annihilation operators.

The creation operator c : H End(F0) is most easily defined by its action on eachSnHn. For f H, define

c(f)h1 h2 hn = 1n

Sn+1(f h1 h2 hn) Sn+1H(n+1)

which we can extend by linearity to F0. The annihilation operator a : H End(F0) issimilarly defined. For f H, define

a(f)h1 h2 hn =

n + 1f, h1h2 hn Sn1H(n1)

which we can similarly extend to F0.Note that since each SnHn is symmetric, c(f) does not need another symmetric projec-

tion in its output. We can heuristically think of c(f) as creating a particle at the support of

f, and a(f) as annihilating one. If the support of f is very small, this models the classical

notion of changing the value of a field at a point. Also note that c is complex linear, but a is

complex anti-linear. In addition, for a given f

H, c(f) and a(f) are adjoint to each other.

Next we can define the Segal field operator S : H End(F0) by

S(f) =1

2(a(f) + c(f)).

Since c(f) and a(f) are adjoint, S(f) is a self-adjoint operator on F, but the mappingf S(f) is not complex linear. This mapping ofH to operators on Fis called the Segal

quantization of H. This construction has some very useful properties illustrated by thefollowing theorem which is thoroughly treated in [20].

Theorem 1.2.1. The Segal quantization S satisfies the following.

i) (Continuity) For all F0, the map f S(f) is continuous

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ii) (Self-adjointness) For all f H, S(f) is essentially self-adjoint

iii) (Cyclicity) F is generated by vectors of the formS(f1)S(f2)

S(fn), where f1,

, fn

Hiv) (Equivariance) If U is a unitary operator on H, then for all f H and F0,

E(U)S(f)E(U)1 = S(Uf)

v) (Commutation Relations) For any f, g H and F0,

[S(f), S(g)] = iIm(f, g)

Segal quantization has many more interesting properties, but these are the ones that

will be of use in this thesis. For more information, see [20]. We can now use the Segal

quantization method to construct an example of a quantum field in the Wightman axioms.

1.3 Example: Free Massless Scalar Quantum Field on

2-D Minkowski Space

As an example we will now construct a free massless scalar quantum field satisfying the

Wightman axioms in two dimensions. This will demonstrate the structure of a quantum

field as well as a common method of construction through Segal quantization.

We begin by constructing the one particle Hilbert space. Define the forward light cone

C as

C = {(E, p) R2|E > 0, E2 p2 = 0}

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and let

D = C0 (C),

the compactly supported smooth functions on C. We can view

Das the space of functions

in C0 (R2) restricted to C, which we can write as

D = {f(|p|, p) | f S(R2)}.

We can put an inner product on D using this restriction by

f, g

= f(|p|, p)g(|p|, p)dp.

The one particle Hilbert space will then be the Hilbert space closure in this inner product

H = (D, , ).

We can then take Fand F0 to be the symmetric Fock space and finite particle space on H.To construct the unitary representation U :

P Aut(

F), we can begin by defining what

happens on H. Writing an element ofPas (a, ) where a R2 and L0, define

U0(a, )h(|p|, p) = ei(|p|a0pa1)h(1(|p|, p)).

Since the translation a corresponds to a phase factor, we need only be concerned with

1(|p|, p) to see that U0 actually maps H to H. Since L0 consists of the Lorentz trans-

formations that preserve both orientation and time direction, and preserves the condition

E2 p2 = 0, we see that for any h H, U0(a, )h H. To see that U0 is unitary, notethat the volume measure on M with respect to the metric is still the standard Lebesgue

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measure. Thus

U(g)h, U(g)h =

|h 1(|p|, p)|2dp =

|h(|p|, p)|2dp = h, h.

We can then extend U0 to Fby E.

U(g)h1 h2 hn = E(U0(g))h1 h2 hn= U0(g)h1 U0(g)h2 U0(g)hn

which remains unitary by the extension of the inner product from H to F.We will also define the operator-valued distribution by first defining a map on the one

particle Hilbert space H. Define 0 : C0 (M) H by

0(f)(|p|, p) =

f(t, x)ei(Etpx)dtdx|E=|p| =

f(t, x)ei(|p|tpx)dtdx

which is the Lorentzian Fourier transform restricted to C. Then clearly 0 is linear and con-

tinuous by properties of the Fourier transform and restriction. We then define the operator

valued distribution using the Segal field operator. If f is real-valued, define

(f) = S(0(f))

and iff = u + iv,

(u + iv) = (u) + i(v).

By Theorem 1.2.1 (i), is an operator-valued distribution.

Now to check the axioms.

Proposition 1.3.1. The quadruple {, U, F, F0} satisfies the Wightman axioms,

W1: (Symmetry) 1, (f)2 = (f)1, 2 for all f C0 (M) and 1, 2 D.

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In particular, if f is real, (f) is self-adjoint, and hence real spectrum

W2: (Vacuum Cyclicity) There is a U-invariant vector F0 with |||| = 1 suchthat the linear span of vectors of the form (f1)

(fn) for fj

C0 (M), n

N is

dense inF0

W3: (Equivariance) U(g)(f)U(g)1 = (f g) for all f C0 (M) and g P

W4: (Causality Commutativity) (f1)(f2) = (f2)(f1) if supp(f1) and supp(f2)

are space-like separated

W5: (Spectral Condition) The infinitesimal generator H of U(t), the time translation

subgroup, has non-negative spectrum.

Proof. Axiom W1 is satisfied by Theorem 1.2.1 (ii) and the linearity of .

For axiom W2, taking = (1, 0, 0, 0, ) F0 as in the previous section, by Theorem1.1 (iii) we need only make sure that 0 reaches a dense subspace ofD, but this comes fromthe fact that the Fourier transform is bijective from S(R2) to itself, and C0 (R2) S(R2) isdense.

To check W3, we compute

0(f (a, )1) =

f((a, )1(t, x))ei(|p|tpx)dtdx

=

f(t, x)ei((|p|,p),(a,)(t,x))dtdx

=

f(t, x)ei((a,)

1(|p|,p),(t,x))dtdx

= ei|p|a0(f)(1(

|p

|, p)).

By Theorem 1.2.1 (iv), we see that is equivariant with respect to U.

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For W4, by Theorem 1.2.1 (iv) we see that

[(f), (g)] = iIm0(f), 0(g)

where the last inner product is on H. Suppose supp(f) and supp(g) are space-like separated.By linearity, we can assume that f and g are real valued. Then

Im(0(f), 0(g)H) =

f(t0, 0)ei(|p|t0px0)g(t1, 1)ei(|p|t1px1)dt0dx0dt1dx1dp

f(t0, 0)ei(|p|t0px0)g(t1, 1)ei(|p|t1px1)dt0dx0dt1dx1dp

= f(t0, 0)g(t1, 1) ei(|p|(t0t1)p(x0x1)) ei(|p|(t0t1)p(x1x1)) dt0dx0dt1dx1dp

=

f(t0, 0)g(t1, 1)(+(t0 t1, x0 x1) +((t0 t1), (x0 x1))dt0dx0dt1dx1

where the distribution + is defined by

+(t, x) =

ei(|p|tpx)dp

By Theorem IX.48 of [20], for t2

x2 < 0,

+(t, x) = fs(t2 x2)

for some fs C(C). Thus

+(t0 t1, x0 x1) +((t0 t1), (x0 x1) = 0

if (t0, x0) and (t1, x1) are space-like separated. Therefore,

Im(0(f), 0(g)) = 0

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which implies that

[(f), (g)] = 0.

To show the spectral axiom W5, recall that time translation is given by U((t, 0), I). The

infinitesimal generator is then given by

Hh(|p|, p) = i ddt

|t=0ei|p|th(|p|, p) = |p|h(|p|, p).

for all h H on which it is defined, and is self-adjoint by Stones theorem. Thus we see thatfor any h D(H),

h,Hh

= |p||h(|p|, p)|2dp 0.

Thus H is positive semi-definite self-adjoint operator, and hence has a non-negative spec-

trum.

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Chapter 2

Axiomatic Quantum Fields on the de

Sitter Surface

2.1 The de Sitter Surface and its Axioms

The de Sitter surface of radius r is defined to be the one-sheeted hyperboloid

SR = {x R3

| x2

0 x2

1 x2

2 = r2

}.

It is of interest to physics as a vacuum solution to the Einstein equations in two dimensions,

and can be used as a two-dimensional model of the early universe.

We can think of Sr as the Lorentzian version of the two-sphere of radius r in R3. In true

mathematical style, we will take r = 1 and call it simply S, since this will not change the

structure of any of the following work.

The isometry group of S is the Lorentz group O(1, 2) on three dimensional Minkowski

space. The space S is a symmetric pseudo-Riemannian space [8] and can be written

S = O(1, 2)/O(1, 1). Of particular interest is the fact that the identity component L0

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O(1, 2) acts transitively on the points of S. In other words, for any x, y S, there is ag L0 so that y = gx.

In addition, a key property of the de Sitter surface is that it admits no global time-like

isometry [15, 14]. Any isometry that acts in a time-like way on part ofS acts in a space-like

way on another part. This fact is the prominent difficulty in any attempts to extend the

spectral axiom of the Wightman axioms to S. Recall that on Minkowski space, the spectral

axiom states that the infinitesimal generator of time translation on the Hilbert space of

quantum states must have a non-negative spectrum. Since de Sitter space admits no global

time translation, this axiom cannot be stated in the same way. Soon we will see that we can

instead get an analogous local condition for S.

We will take coordinates for S as follows.

(t, )

tan t

sec t cos

sec t sin

for t (

2 ,

2 ) and [, ). Note that tan2

t sec2

t cos

2

sec2

tsin

2

= 1. In thesecoordinates, the Lorentz metric induced from R3 takes the form

= sec2 t

1 0

0 1

.

The key property of this coordinate metric is that it is conformally flat. The dAlembertian

operator then takes the form

= cos2 t

2

t2

2

2

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which we will use in the construction of a quantum field on S.

The volume measure on S is given by

dVol(t, ) = sec

2

t dtd.

Since the metric, and hence causal structure, of S is induced from that ofR3, in these

coordinates we see that for v, w S, w is in the light cone of v if

0 = v w, v w = v, v 2v, w + w, w = 2 2v, w.

Thus w is in the light cone of v if

v, w = 1

In coordinates v = (t, ) and w = (t0, 0) we see

tan t tan t0 sec t sec t0 cos cos 0 sec t sec t0 sin sin 0 = 1.

Multiplying through by cos t cos t0, which is positive in our range of t and t0, we get

sin t sin t0 cos cos 0 sin sin 0 = cos t cos t0

which finally gives the equation

cos(t t0) = cos( 0).

Again, given the range of t on S, this implies that the light cone boundary at v = (t, ) on

S is given by the equations

(t t0) + ( 0) = 0

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(t t0) ( 0) = 0

or better stated as

(t

t0)

2

(

0)

2 = 0

superficially matching Minkowski space. This is not unexpected given that metric in these

coordinates is conformally flat. Thus two points of S are space-like separated if t2 2 < 0,time-like separated if t2 2 > 0, and light-like separated if t2 2 = 0 as in Minkowskispace.

We can also view the homogeneous Lorentz group L0 in these coordinates. Recall that

L0 is generated by three one-paramter groups

B1() =

cosh sinh 0

sinh cosh 0

0 0 1

B2() =

cosh 0 sinh

0 1 0

sinh 0 cosh

R() =

1 0 0

0 cos sin 0 sin cos

.

In (t, ) coordinates, this gives

B1()(t, ) =

tan1(cosh tan t + sinh sec t cos ),

tan1(sec t sin

sinh tan t + cosh sec t cos )

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B2()(t, ) =

tan1(cosh tan t + sinh sec t sin ),

tan1(sinh tan t + cosh sec t sin

sec t cos )

R()(t, ) = (t, + ).

Similar to Minkowski space, we will define an operator valued distribution on S to

be a linear map

: C0 (S) End(F0)

where F0 Fis a dense linear submanifold of a Hilbert space Fsuch that for all 1, 2 F,the map

f 1, (f)2

is a continuous linear functional using the Schwartz topology on C0 (S).

We would like to define a scalar quantum field on S as a quadruple {, U, F, F0},where F is a separable Hilbert space, F0 a dense linear submanifold,

: C0 (S) End(F0)

an operator valued distribution and

U : L0 Aut(F)

a unitary representation, which satisfies axioms similar in spirit to the Wightman axioms.

This provides difficulties as shown in [16, 3, 4]. The first three axioms generalize quite

naturally to S.

CW 1: (Conjugate Hermiticity) For all f C0 (S) and all 1, 2 F0,1, (f)2 = (f)1, 2.

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CW 2: (Vacuum Cyclicity) There exists a vector F0 with |||| = 1 such thatthe linear span of vectors of the form (f1) (fn) is dense in F0.

CW 3: (Equivariance) For all g

L0 and f

C0 (S),

U(g)(f)U(g)1 = (f g1).

We would like a version of the causal commutativity axiom on S, such as

CW 4: (Causal Commutativity) If f1, f2 C0 (S) and supp(f1) and supp(f2) arespace-like separated, then the commutator [(f1), (f2)] = 0.

The analysis of an axiom of this type, however, will be reserved for a future work.

The Minkowski space positivity axiom W5, which states that the representation U must

give a generator of time translation that has non-negative spectrum, is more difficult. On

Minkowski space, time translation is a global time-like isometry, so this requirement makes

sense. S, however, has no global time-like isometry. Our only hope is to look locally.

CW 5: (Local Positivity) About each x S there is a neighborhood W, a oneparameter group G()

L0, and a corresponding subspace

FW

Fsatisfying the

following. Let : F FW be orthogonal projection. The infinitesimal generator Aof U(G()) U(L0) projected to FW as A is a non-negative self-adjoint operator,and the Killing field corresponding to G() is time-like on W.

With this axiom, we are guaranteed a local representation of time translation, U(G()), and

its infinitesimal generator A will act as a local Hamiltonian on W.

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2.2 Example: Free Massless Scalar Quantum Field on

the de Sitter Surface

In order to demonstrate the consistency of these axioms and get a feel for their structure, we

will now construct an example of such a scalar quantum field on S. To accomplish this, we

must construct the quadruple {, U, F, F0} discussed in the previous section. Our examplewill be constructed by Segal quantization, so we begin by defining a one particle Hilbert space

H with a dense linear submanifold D. For notational ease, write n = ein|n| for n Z {0},and let D = span({n}nZ{0}). Define an inner product , on D by

n, m = (n m)

for all n, m Z {0}, and extending by complex linearity. Let H be the completion of Din this inner product. Then

H = nZ{0}

ann |nZ{0}

|an|2 <

.

Note that in this inner product, H = H+ H where

H+ = span({n | n > 0})

H = span({n | n < 0}).

We can write

f1(), f2() = i2

f1()

d

df2()d

where dd

denotes dd

on H+ and dd

on H.Now to construct the Hilbert space F by Segal quantization as we did on Minkowski

20


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space. Build Fand F0 as the symmetric Fock space and finite particle space on H. Take

F0 = C (n=1SymnHn)

and the Hilbert space completion F= F0.Now we build the field map : C0 (S) End(F0) also by Segal quantization. Inspired

by de Bievre and Renaud [3, 4], we define 0 : C0 (S) H by

0(f)() =nN

f(t0, 0)

1n

ein(t0+0) dVol(t0, 0)1n

ein

+ nN f(t0, 0)

1

nein(t00) dVol(t0, 0)

1

nein.

Intuitively, the purpose of this map is to propagate the support of f out into forward and

backward wave equation solutions while compensating for the geometry of S, but only keep

the positive frequency in t parts, and then reduce to t = 0 for the Cauchy data. We can

write 0(f) more concisely as

0(f)() = nZ{0}

f(t0, 0) 1|n|ei|n|t0in0 dVol(t0, 0)n

To show that this map does actually make sense, we will compare with the functions

+(f)() =

f(t0, t0)sec2 t0dt0, (f)() =

f(t0, + t0)sec

2 t0dt0.

First note that since f is smooth and has compact support, (f) C(S1). Indeed,

dk

dk

f(t0, t0)sec2 t0dt0 =

f(0,k)(t0, t0)sec2 t0dt0

where f(0,k) represents the kth order partial derivative off on the second variable. Since f(0,k)

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valued f C0 (S),(f) = S(0(f))

and for f = u + iv,

(f) = (u) + i(v)

Proposition 2.2.1. is an operator valued distribution.

Proof. Suppose fk f in the Schwartz topology ([19] pg. 133) for a sequence fk C0 (S).Then in particular, fk f uniformly on any compact set. Without loss of generality, wecan assume there is a compact set K S containing the supports of all fk and f. Thus

fk(t0, 0)

1n

ei(|n|t0+n0) dVol(t0, 0)

f(t0, 0)1n

ei(|n|t0+n0) dVol(t0, 0)

for each n Z{0}, and hence 0(fn) 0(f) in H. Therefore 0 is continuous. Theorem1.2.1 (i) then gives that f 1, S(0(f))2 is continuous by composition.

Last, we will construct the representation U : L0 Aut(F). First, we will construct a

representation U0 : L0 Aut(H), and then extend to Fby E.

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Define three operators on H by the following. For h =

nZ{0}hnn H,

A1h =

2

2

h12 +

n=2

hn

2n(n + 1)n+1 +n(n

1)n1+

2

2h12 +

n=2

hn2

n(n + 1)(n+1) +

n(n 1)

(n1)

A2h =i

2

2h12 +

n=2

hn2

i

n(n + 1)n+1 i

n(n 1)

n1

+ i

2

2h12 +

n=2

hn2

i

n(n + 1)(n+1) i

n(n 1)

(n1)

A3h = nZ{0}

nhnn

when the result is defined. We can write this more concisely as

A1 = i cos

A2 = i sin

A3 = i

with the understanding that constant terms are mapped to zero.

Lemma 2.2.1. A1, A2, and A3 are essentially self-adjoint.

Proof. We can show that these operators are self-adjoint by viewing them as matrix operators

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in the ordered basis = {1, 1, 2, 2, }. Note that

n, A1m = n, i cos (i|m|m)

= n,1

2|m(m + 1)|m+1 +

1

2|m(m 1)|m1

=1

2

|m(m + 1)|0(m + 1 n) + 1

2

|m(m 1)|0(m 1 n)

n, A2m = n, i sin (i|m|m)

= n, i2

|m(m + 1)|m+1 + i

2

|m(m 1)|m1

=i2

|m(m + 1)|0(m + 1 m) + i

2

|m(m 1)|0(m 1 n)

n, A3m = n, i(im)m

= m0(m n)

which gives infinite dimensional matrices

A1

=

0 0 12

0 0 0 0 0 1

20

12

0 0 062

0 12

0 0 0 0 0

62

A1 =

0 0 i2

0 0 0 0 0 i

20

i

2 0 0 0 i6

2 0 i

20 0 0

0 0 i6

2

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A3 =

1 0 0 0 0 1 0 0 0 0 2 0

0 0 0 2

Clearly each is Hermitian symmetric, and by Theorem 4 of page 104 of [1], since the columns

of each are square summable, each defines a closed operator.

Recall that a vector v H is called analytic for an operator A if there is a t > 0 suchthat

k=0

||Akv

||k! tk

< .We will now show that each n is an analytic vector for all three Ai operators. A straight-

forward calculation shows that

||Ak1n|| 2k

n

n + k(n + 1)(n + 2) (n + k)

noting that the highest index basis vector in Ak1n is n+k which attains a multiple of

n

n + k(n + 1)(n + 2) (n + k). This is the largest coefficient expanding Ak1n in thebasis, and applying the triangle inequality, the above result holds. Taking t = 12n! , we see

that

||Ak1n||tk 1

n!

n

n + k(n + 1)(n + 2) (n + k) 1.

Therefore,

k=0

||Akn||k!

tk

k=0

1

k!

= e.

Thus n is an analytic vector for A1. A similar calculation shows that n is an analytic vector

for A2 as well, and a comparatively trivial calculation gives the same result for A3. Therefore

each Ai has a total set of analytic vectors in its domain. By Theorem X.39 (Nelsons analytic

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vector theorem) of [20], we see that each Ai is essentially self-adjoint.

From this point on, we will replace each Ai with its unique self-adjoint extension. By

Theorem VIII.7 of [19], if A is a self-adjoint operator, then eiA forms a stronglycontinuous one-paramter unitary group. Recall that the basis elements Li of the Lie algebra

L0 ofL0 satisfy the commutation relations

[L1, L2] = L3, [L1, L3] = L2, [L2, L3] = L1.

Also, the Ai satisfy

[iA1, iA2] = iA3, [iA1, iA3] = iA2, [iA2, iA3] = iA1

showing that Li iAi is a Lie algebra isomorphism.By a result of Nishikawa [17], we have the fact that the exponential map exp : L0 L0

is surjective. Viewing L0 as the matrix group on R3, we can take exp to be the exponential

series. We will define the unitary representation U0 : L0 Aut(H) by

U0(eaL1+bL2+cL3) = eiaA1ibA2icA3.

In order to show that this map is well-defined, we will first need some tools. First we will

define three one-parameter unitary representations

U(B1()) = eiA1,

U(B2()) = eiA2,

U(R()) = eiA3.

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Note that B1() and B2() are one-to-one, and eiA3 is translation by , which has period2 since H consists of functions on S1. Thus each of these is well-defined.

We begin by pointing out that any element ofL0, the Lie algebra ofL0 can be written as

L =

0 a b

a 0 cb c 0

with a,b,c R. Also, we see that the eigenvalues ofL are {0, a2 + b2 c2, a2 + b2 c2}.

Lemma 2.2.2. Suppose L, L L0 take the forms

L =

0 0 0

0 0 c0 c 0

, L =

0 a b

a 0 c

b c 0

and eL = eL

. Then a = b = 0. In particular, if eL = I, then

L =

0 0 00 0 2n0 2n 0

for some n Z.

Proof. Suppose L and L satisfy the hypotheses above. Let

T =

n 0 0

0 1 0

0 0 1

.

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We see that for any n > 0, T is invertible. Thus eTLT1

= eTLT1 for any n > 0. Notice

that

T LT1 =

0 0 0

0 0

c

0 c 0

, T LT1 =

0 na nb

na

0

c

nb c 0

.

Thus the eigenvalues of T LT1 are still

{0,ic, ic},

but the eigenvalues of T LT1 are

{0,

n2(a2 + b2) c2,

n2(a2 + b2) c2}.

Hence the eigenvalues of the respective exponentials are

{1, eic, eic}, {1, e

n2(a2+b2)c2, e

n2(a2+b2)c2}.

We may choose n large enough that

n2(a2 + b2) c2 is real and greater than 1. Theneic cannot equal e

n2(a2+b2)c2 or e

n2(a2+b2)c2. Thus eTLT

1 = eTLT1, which is acontradiction. Therefore it must be the case that a = b = 0.

Lemma 2.2.3. Suppose L, L L0 each have real distinct eigenvalues and eL = eL. ThenL = L.

Proof. Suppose L, L each have distinct real eigenvalues. Then there exists a > 0 such

that L + I and L + I each have distinct positive eigenvalues. Thus we can take the matrix

logarithm, seeing

ln(eL+I) = L + I = ln(eL+I) = L + I

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which implies that L = L.

Now we can prove the following.

Proposition 2.2.2. The map U0 :

L0

Aut(

H) is well-defined.

Proof. Suppose L, L L0 and eL = eL . Write

L =

0 a b

a 0 cb c 0

, L =

0 a b

a 0 c

b c 0

.

Case 1 : Suppose the nonzero eigenvalues ofL are imaginary. Then in particular,|c|

>|a|

and |c| > |b|. Conjugating L by B1(), we see that

B1()LB1() =

0 a b cosh c sinh a 0 b sinh c cosh

b cosh c sinh b sinh + c cosh 0

.

Since

|bc

|< 1, we can solve b cosh

c sinh = 0. Thus we may write

B1()LB1() =

0 a 0

a 0 c0 c 0

.

Since conjugating by B1() cannot turn the imaginary eigenvalues real, we see that |a| < |c|.Conjugating by B2(), we see that B2()B1()LB1(

)B2(

)

=

0 a cosh c sinh 0a cosh c sinh 0 a sinh c cosh

0 a sinh + c cosh 0

.

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Since |ac| < 1, we can solve a cosh c sinh = 0. Thus we can write

B2()B1()LB1(

)B2(

) =

0 0 0

0 0

c

0 c 0

.

By Lemma 2.2.2, then, we see that

B2()B1()LB1()B2() =

0 0 0

0 0 c

0 c 0

.

for some c R. Thus

eB2()B1()LB1()B2() = R(c) = R(c) = eB2()B1()LB1()B2(),

which implies that

B2()B1()eL

B1()B2()= B2()B1()e

LB1()B2(),

which means

U0(B2())U0(B1())U0(eL)U0(B1())U0(B2())

= U0(B2())U0(B1())U0(eL)U0(B1())U0(B2()).

Conjugating back out, we see that

U0(eL) = U0(e

L).

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Case 2: Suppose the eigenvalues ofL are real. If the eigenvalues ofL are also real, then

either each has distinct real eigenvalues, or all eigenvalues are zero. In the first case, by

Lemma 2.2.3, L = L. In the second case, by Lemma 2.2.2, eL = eL

= R(c). In either case,

U0(eL) = U0(eL

).

If the nonzero eigenvalues of L are imaginary, then we may apply Case 1 reversing the

roles of L and L. Therefore we see that U0 is well-defined.

Since the Ai have a common dense set of analytic vectors in their domains, we may

view the exponential of any aA1 + bA2 + cA3 as its exponential series. Thus U0 is also a

homorphism, and hence a unitary representation ofL0 on H.

Extend U to Fby applying the map E from Segal quantization, U(g) = E(U0(g)). Inother words,

U(g)h1 h2 hn = U0(g)h1 U0(g)h2 U0(g)hn

extended by linearity. Thus we have our unitary representation U : L0 Aut(F). Thequadruple {, U, F, F0} will be our quantum field on S.

Now we may justify the adjectives free and massless for this field. Since

2

t20

2

20

ein(t00) = 0

we see that

0(f) = nZ{0}f(t0, 0)

1

|n

|

ei|n|t0+in0 dVol(t0, 0)n

=

nZ{0}

cos2 t0

2

t20

2

20

(f(t0, 0))

1|n|ei|n|t0+in0 sec2 t0dt0d0n=

nZ{0}

f(t0, 0)

1|n|

2

t20

2

20

ei|n|t0+in0dt0d0n = 0.

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2.3 Axioms CW1-CW3

Now we will check that the quantum field above satisfies the axioms CW1-CW3. As men-

tioned previously, the form and analysis of a CW4 axiom for causal commutativity will be

reserved for a future work. The spectral axiom, CW5 is involved enough to deserve a separate

treatment in the following sections.

Proposition 2.3.1. The field{, U, F, F0} satisfies the following CW axioms for a quantumfield onS.

CW 1: (Conjugate Hermiticity) For all f C0 (S) and all 1, 2 D,

1, (f)2 = (f)1, 2

CW 2: (Vacuum Cyclicity) There exists a vector F0 with |||| = 1 such thatthe linear span of vectors of the form (f1) (fn) is dense in F0.

CW 3: (Equivariance) For all g L0 and f C0 (S),U(g)(f)U(g)1 = (f g1)

Proof. CW1 is satsified by Theorem 1.2.1 (ii) of Segal quantization and the complex linearity

of .

To show CW2, first recall that we can approximate a delta function on (2

, 2

) by a

sequence of real functions in C((2

, 2

)) with increasingly smaller support around zero.

In other words, there exists a sequence {hk(t)} C((2 , 2 )) such that for all f C((2 , 2 )),

limkhk(t)f(t)dt = 0(t)f(t)dt = f(0)

Suppose g =

nZ{0}gnn H, and g is smooth. Define g =

nZ{0}

|n|2

gnn, and pn(t, ) =

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hn(t)g(). Then

limk

0(pk)() = limk

nN

hk(t0)

|n|

2g(0)e

in(t0+0) dVol(t0, 0)n

+nN

hk(t0)

|n|2

g(0)ein(t00) dVol(t0, 0)n

=nN

0(t0)

|n|2

g(0)ein(t0+0) dVol(t0, 0)n

+nN

0(t0)

|n|2

g(0)ein(t00) dVol(t0, 0)n

=

nZ{0}i2

g(0)

0nd0n

=

nZ{0}n, gn = g.

Viewing H as a set in L2(S1), we see that C(S1)H is dense. Thus we can approximate anyg H, which means 0(C0 (S)) = H. Then by Theorem 1.2.1 (iii), taking = (1, 0, 0 ) F0, we have that the linear span of vectors of the form (f1) (fn) is dense in F0.

For CW3, first note that since U is defined in terms of infinitesimal generators, we need

to check the three equations

id

d|=00(f B1()1) = A10(f)

id

d|=00(f B2()1) = A20(f)

id

d|=00(f R()1) = A30(f).

Let h(t, ) = t . We will abuse notation a bit and write h g(t, ) as g(t ). Using

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the fact that the volume measure is invariant under L0, we calculate

id

d|=00(f B1()1)

= id

d |=0nN

f B1()

1

(t0, 0)1

nein(t0+0)

dVol(t0, 0)ein

+ id

d|=0

nN

f B1()1(t0, 0)ein(t00) dVol(t0, 0)ein

= id

d|=0

nN

f(t0, 0)

1n

einB1()(t0+0) dVol(t0, 0)ein

+ id

d|=0

nN

f(t0, 0)

1n

einB1()(t00) dVol(t0, 0)ein

= inN

f(t0, 0) 1n cos(t0 + 0)

t0ein(t0+0) dVol(t0, 0)

1

nein

+ inN

f(t0, 0)

1n

cos(t0 0) t0

ein(t00) dVol(t0, 0)1n

ein

=nN

f(t0, 0)

n

2

ei(n+1)(t0+0) + ei(n1)(t0+0)

dVol(t0, 0)

1n

ein

+nN

f(t0, 0)

n

2

ei(n+1)(t00) + ei(n1)(t00)

dVol(t0, 0)

1n

ein

Reindexing, we get

m=2

m 1m

f(t0, 0)

1m

eim(t0+0) dVol(t0, 0)1

m 1ei(m1)

+

m=0

m + 1

m

f(t0, 0)

1m

eim(t0+0) dVol(t0, 0)1

m + 1ei(m+1)

+

m=2

m 1m

f(t0, 0)

1m

eim(t00) dVol(t0, 0)1

m 1ei(m1)

+

m=0

m + 1mf(t0, 0) 1meim(t00) dVol(t0, 0) 1m + 1 ei(m+1)= A10(f).

Similar calculations show the necessary results for A2 and A3.

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The last axiom, CW 5, is involved enough to deserve its own section following the next

preliminary result.

2.4 Diagonal Dominance

Our examination of the local spectral axiom CW 5 will require the following results about a

special class of operators.

Let A : H H be a symmetric linear operator with dense domain D(A) on a separableHilbert space

Hwhich is not necessarily bounded, and suppose =

{e1, e

2, } D

(A) be

an orthonormal basis for H. For notational purposes, label Anm = en, Aem for m, n N.Then A is diagonally dominant with respect to if

|Ann| m=n

|Anm|

for all n. A is strictly diagonally dominant if the inequality is strict. We will not

explicitly indicate the basis if it is clear. Note that since A is symmetric, there is no need

to specify row or column diagonal dominance. I will refer to the set {Ann} as the diagonal.The following are some results about diagonally dominant operators that will be useful

in the examination of CW5.

Lemma 2.4.1. Suppose A is strictly diagonally dominant and self-adjoint. Then ker(A) =

{0}.

Proof. Suppose A is strictly diagonally dominant and self-adjoint, and suppose Ax = 0 for

some x = 0 D(A). Then en, Ax = 0 for all en . Write x =

m=1

xmem, and choose k

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so that |xk| is maximal. Then

|ek, Ax| = |

m=1

Aknxn| = |Akkxk +n=k

Aknxn|

|Akkxk| n=k

|Aknxn| |Akkxk| n=k

|Aknxk|

by the maximality of |xk|. Using strict diagonal dominance, we get the strict inequality

|xk|(|Akk| n=k

|Akn|) > |xk|(|Akk| |Akk|) = 0

which is a contradiction.

The following result will allow us to retreat to a smaller domain for a self-adjoint A.

Lemma 2.4.2. Suppose A is symmetric and diagonally dominant with real diagonal. Then

A is essentially self-adjoint.

Proof. By the corollary of Theorem VIII.3 of [19], a symmetric A is essentially self-adjoint if

and only if ker(A

iI) =

{0}

. Note that since A is symmetric and densely defined, A is a

self-adjoint operator with the same matrix representation. Thus A is diagonally dominant

with real diagonal.

Since Ann R, |Ann i| > |Ann|. Thus A iI is strictly diagonally dominant. ByLemma 2.4.1, ker(A iI) = {0}.

This leads us to the following.

Proposition 2.4.1. Suppose A is self-adjoint and diagonally dominant with positive real

diagonal. Then A is positive semi-definite.

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Proof. Suppose A satisfies the hypotheses above. Retreat to the dense domain

D(A) = {x H | x =N

n=1

xnen f or some N N},

the set of finite sum vectors. Then A

= A|D(A) defines a symmetric diagonally dominantoperator with real diagonal. By Lemma 2.4.2, A

is essentially self-adjoint, and hence has a

unique self-adjoint extension (namely A).

Now, note that for any x D(A), x, Ax 0 by the positivity of finite dimensionaldiagonally dominant matrices with positive diagonal [13].

By Theorem X.23 of [20] (Friedrichs extension), A

has a positive semi-definite self-adjoint

extension, but since A

is essentially self-adjoint, A is the only self-adjoint extension of A

.

Thus A is positive semi-definite.

2.5 CW5, the Local Spectral Condition

Now we will check that the local spectral condition holds for this field.

Proposition 2.5.1. The field {, U, F, F0} satisfies the axiom CW 5 (Local Positivity).About each x S there is a neighborhood W, a one parameter group G() L0, and acorresponding subspace FW F satisfying the following. Let : F FW be orthogonalprojection. The infinitesimal generator A of U(G()) U(L0) projected to FW as A isa non-negative self-adjoint operator, and the Killing field corresponding to G() is time-like

on W.

Proof. Begin by choosing the base point x0 = (0, 0) in (t, ) coordinates. Our candidate for

a locally positive operator for x0 will be the element A = A1, with A1 the infinitesimal

generator of U(B1()), and : H HW the projection. First we will find a suitable Wcontaining x0, and then construct a local one particle subspace HW. Next we will check that

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A has non-negative spectrum on HW using the diagonal dominance results above. Then wecan construct a local Fock subspace FW Fand still have a non negative spectrum whenE(A) is restricted to it. Last we will use the homogeneity of the de Sitter space S to get

locally positive operators on such neighborhoods for any other point in S.

Take W S to be the neighborhood

W =

(t, )|t2 2 < 4

.

Define HW by

HW = (4 ,4 )ei4n

|n|nZ{0}where the bar signifies the completion as a subspace of H. For notational purposes, taken = (4 ,4 )

ei4n|n| . We will order this basis as = {

1,

1,

2,

2, }. We can now view

A as a matrix operator with the matrix elements

Anm = An, m =i2

4

4

icos()i4|n|ei4n|n| i4|m|

ei4m|m|d

=8

2|mn|

4

4

ei(4m4n1)ei(4m4n+1)d

= 8

2|mn|

cos((m n))16(m n)2 1

Note the symmetry in m and n, showing that A a symmetric operator. Another important

feature of these matrix elements is that the diagonal takes the form

Ann = 82

|n|

with is positive and relatively large.

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We will now show that the matrix operator A is diagonally dominant with respect to .

Note

m=n|Anm| = 8

2

m=n

|nm|16(m n)2 1

8

2|n|

m=n

|m|

15(m n)2

since m, n > 0 and m = n. Replacing k = m n, we see this is

8

2|n|15

kZ

|k + n|

k2 8

2|n|

152kN

k

k2+

|n|

k2

= 8

2|n|15

(2(32

) +|n|2

3) < 82

|n|

To see that A is self-adjoint, note that for a fixed n,

mZ{0}

|Anm|2 = 322

(|n|2 +m=n

|nm|(16(m n)2 1)2 )

32

2|n|

2 +|n|m=n

|m|225(m n)4

322

|n|2 + |n|

k=0

|k + n|225k4

322

|n|2 + 2|n|

k>0

1

225k3+

|n|225k4

<

By Theorem 4 of page 104 of [1], since the columns of A are square summable, A is a closed

operator.

By Proposition 2.4.1, since A is a self-adjoint diagonally dominant operator with positive

real diagonal, A is a positive semi-definite operator on HW. Thus A has a non-negativespectrum.

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Now we can construct FW F as the Fock space built from HW. Then E(A) will bedefined by

E(A)(h1 h2 hn) = Ah1 Ah2 Ahn

extended by linearity to F. Thus we see that E(A) has a non-negative spectrum.Next we check that the Killing field corresponding to B1() is time-like on W. Since the

causal structure ofS is inherited from its embedding in three dimensional Minkowski space,

we can compute the Killing field there by

v =d

d=0

cosh() sinh() 0

sinh() cosh() 0

0 0 1

tan(t)

sec(t)cos

sec(t)sin

=

sec(t)cos

tan(t)

0

and see that

v, v = cos2() sin2(t).

The map (t, ) cos2() sin2(t) is continuous, and cos2() sin2(t) = 0 when = 2 tand = 2 + t. Since cos

2(0) sin2(0) = 1, we see by continuity that v, v > 0 on W istime-like.

Using the fact that L0 acts transitively on S, for any x S, there is a g L0 such thatx = gx0. To satisfy the required conditions at x, we take

W = gW, A = U(g)AU(g)1, B() = gB1()g1,

HW = U(g)H

W,

FW = U(g)

FW

By the equivariance axiom CW3, A is the infinitesimal generator ofB, and A = U(g)AU(g)1.

Since A is defined as a unitary transformation of A, A also has a non-negative spectrum

on HW , giving the local positivity condition at x. In addition, since g is an isometry of S,

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the Killing field corresponding to B() is time-like on W. Thus we can satisfy the required

conditions of CW5 at any point x S.

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Chapter 3

Connections and Speculations

3.1 Connections to Other Work in the Field

The topics of quantum fields on de Sitter space and infinite dimensional diagonally dominant

matrix operators are not new, and many authors have contributed to the literature on these

subjects. I will now discuss links between the results of Chapter 2 and some of these previous

works.

The spectral properties of symmetric universally diagonally dominant matrix operators

on a separable Hilbert space have been studied in [7] and [6]. Universal diagonal dominance,

however, is a difficult condition to attain for unbounded operators, and section 2.4 provides

a slight generalization. The equivariance of the field map in section 2.2 would on first glance

seem to contradict de Bievre and Renaud in [3] and [4], but this is not the case. Renaud and

de Bievre point out that the nonconstant modes of the wave equation are not invariant

under the regular representation of

L0 by precomposition. The U in Chapter 2, however, is

not the regular representation, but essentially a projected version of it. There is no claim

that the representation in Chapter 2 meets all physical concerns of de Bievre and Renaud.

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3.2 Speculations

The work presented in this thesis generates many possibilites for extensions and modifica-

tions. The first of these, of course, is the inclusion of a causal commutativity axiom. While

the statement of the Wightman axiom W4 can be easily modified to make sense on the de

Sitter surface,

CW 4: (Causal Commutativity) If f1, f2 C0 (S) and supp(f1) and supp(f2) arespace-like separated, then the commutator [(f1), (f2)] = 0,

the analysis of such an axiom is a bit subtle. The most common method as used in [3] is to

construct the Greens function as a sum of modes and their conjugates. This does not work

when we give up the constant modes, so another approach will have to be taken. While we

did not examine the causal commutativity of the field in Chapter 2, we can present a field

satisfying the above axiom at the cost of equivariance.

If instead of taking 0 as defined previously, define c by simply

c(f) = nN f(t0, 0)e

in(t0+0) dVol(t0, 0)ein

+nN

f(t0, 0)e

in(t00) dVol(t0, 0)ein

or in more concise terms,

c(f) =

nZ{0}

f(t0, 0)e

i|n|t0+n dVol(t0, 0)ein.

To show CW4 for this field, suppose f1, f2 C0 (S) and supp(f1), supp(f2) are space-likeseparated. By Theorem 1.2.1 (v), [(f1), (f2)] = iIm(c(f1), c(f2))H. By linearity, we

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can assume f1 and f2 are real valued. We see that

c(f1), c(f2) =nN

n

f1(t0, 0)f2(t1, 1)e

in(t0+0t11)dV ol(t0, 0)dV ol(t1, 1)

+nN

n

f1(t0, 0)f2(t1, 1)ein(t00t1+1)dV ol(t0, 0)dV ol(t1, 1)

Thus

2iIm(c(f1), c(f2)) =nN

n

f1(t0, 0)f2(t1, 1)e

in(t0+0t11)dV ol(t0, 0)dV ol(t1, 1)

+nN

n

f1(t0, 0)f2(t1, 1)e

in(t00t1+1)dV ol(t0, 0)dV ol(t1, 1)

nN

n

f1(t0, 0)f2(t1, 1)ein(t0+0t11)dV ol(t0, 0)dV ol(t1, 1)

+nN

n

f1(t0, 0)f2(t1, 1)e

in(t00t1+1)dV ol(t0, 0)dV ol(t1, 1)

=

nZ{0}

n(ein(t0+0t11) ein(t00t1+1))f1(t0, 0)f2(t1, 1)dV ol(t0, 0)dV ol(t1, 1)

We will show that this is zero by comparing it to an inner product on L2(S1). Notice for

all (t0, 0) supp(f1) and all (t1, 1) supp(f2),

(t0 t1)2 (0 1)2 < 0

since the supports are space-like separated. Consider the functions

+(f)() =

0( t0 0)f(t0, 0)dV ol(t0, 0)

(f)() =

0( t0 + 0)f(t0, 0)dV ol(t0, 0)

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Suppose is in both supp(+(f1)()) and supp(+(f2()). Then there must exist a (t0, 0) supp(f1) and (t1, 1) supp(f2) so that

t0 0 = t1 1 = 0

Subtracting, we see that then

(t0 t1) + (0 1) = 0

which implies

((t0 t1) + (0 1))((t0 t1) (0 1)) = (t0 t1)2 (0 1)2 = 0

which is a contradiction. Therefore, supp(+(f1)()) and supp(+(f2)()) are disjoint.

Since taking a derivative will not expand the support, this means supp(+(f1)()) and

supp( dd

+(f2)()) are disjoint. A similar argument gives the same result for (f1)() and

(f2)(). Recalling our L2(S1) expansions of (f)(), we see

0 = +(f1)(), dd

+(f2)()L2(S1) + (f1)(), dd

(f2)()L2(S1)

=nZ

f1(t0, 0)e

in(t0+0)eind

d(f2(t0, 0)e

in(t0+0)ein dVol(t0, 0) dVol(t1, 1)d

+ f1(t0, 0)ein(t00)ein

d

d(f2(t0, 0)e

in(t00)ein dVol(t0, 0) dVol(t1, 1)d

=

nZ{0}

inf1(t0, 0)e

in(t0+0)(f2(t1, 1)ein(t1+1) dVol(t0, 0) dVol(t1, 1)

+inf1(t0, 0)ein(t00)f2(t1, 1)ein(t11) dVol(t0, 0) dVol(t1, 1)

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If f1 and f2 are real, then this gives us

0 =

nZ{0}

in(ein(t0+0t11) ein(t00t1+1))f1(t0, 0)f2(t1, 1) dVol(t0, 0) dVol(t1, 1)

= 2Im(c(f1), c(f2)H)

as seen above. Thus [(f1), (f2)] = 0, giving causal commutativity. However, c is not

equivariant with respect to U. Thus we have a causal field map that is not equivariant.

The statements of the CW axioms generalize readily to any symmetric space that admits

a Lorentzian metric in the following way. Suppose M is a Lorentzian symmetric manifold

with isometry group G. Then a quantum field on M can be taken as a quadruple , U,

F,

F0

with the analogous operator-valued distribution and unitary representation U of G0, the

identity component of G. The axioms would then make sense replacing S with M and L0with G0. Due to the symmetric nature ofM, we would then be able to move a local spectral

condition from one basepoint to another in the same manner. The challenge is then to

construct interesting examples satisfying these or closely related axioms on other Lorentzian

symmetric spaces.

Another area to explore is the relationship between a quantum field on de Sitter space

and the corresponding field on Minkowski space. A possible method of considering the

relationship is to construct the analogous field from Chapter 2 on a de Sitter surface of

radius r, and investigate the r limit. I have done some work on this topic, and one cansee that the neighborhood W about the basepoint x0 limits in a sense to the Minkowski plane

and its origin. In addition, the boost B1() becomes time translation, and R() becomes

translation in space. Future work will explore what happens to the map in this limit, and

if it corresponds to the free field map on Minkowski space.

Finally, one would like to explore many of the results of the Wightman axioms in the

context of de Sitter space. It is possible that on the de Sitter surface, some uniqueness

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a local spectral condition for axiomatic quantum fields on the de sitter surface

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