hellman - probability in qm

Click here to load reader

Post on 04-Oct-2015

216 views

Category:

Documents

2 download

Embed Size (px)

DESCRIPTION

Hellman - Probability in QM

TRANSCRIPT

  • Interpretations of Probability inQuantum Mechanics: A Case of"Experimental Metaphysics"

    Georey Hellman

    1 Introduction

    One of the most philosophically important and fruitful ideas emergingfrom the work of Abner Shimony et al. relating to the Bell theorems,named and highlighted by Shimony, is that of "experimental meta-physics". (See e.g. Shimony [1984].) Although the general view thatthere is no sharp boundary between metaphysics and natural scienceand that questions in the former domain are aected by empirical evi-dence bearing directly on the latter is not new, and indeed forms a cen-tral tenet of mid-twentieth-century Quinean philosophy of science, thelinks between experimental tests of Bell inequalities, for example, bearfar more directly on matters standardly called "metaphysical" than eventhat sophisticated philosophy could ever have anticipated. Those linksfor example, between experiments of Aspect, et al.[1982] and theses of"local realism" and "local determinism" stand independently of anyappeal to Duhemian-Quinean "holism" of testing (according to whichit is really whole theories, including various metaphysical backgroundprinciples, that are tested by experimental evidence, rather than in-dividual statements). What is truly extraordinary about the tests ofBell-type inequalities is the directness of the role of the metaphysicaltheses, e.g. Einsteins principle of separability of physical states accord-ing to space-time location, leading to mathematically precise conditionsconstraining assignments of values of relevant quantities of local hid-den variables theories. The accumulated wealth of evidence conrmingquantum correlations between separated subsystems (e.g. paired pho-

    The author is grateful to Wayne Myrvold and Michel Janssen for useful commentsand to audiences at the Seven Pines Symposium (May, 2006) and at the PerimeterInstitute Symposium in Honor of Abner Shimony (June, 2006) for helpful discussion.

    1

  • tons in atomic cascades), thus violating relevant Bell-type inequalities,tells quite directly that aspects of physical nature as we understand itviolate separability. The same holds, mutatis mutandis, with respect tothe conclusion that certain systems exhibit (temporally or dynamically)indeterministic behavior, that certain "actualization" phenomena (e.g.of a value of polarization or spin in a specied orientation) occur "ran-domly", violating the entrenched rationalist principle of causality (or"su cient reason").But what about "loopholes"? Testing a Bell-type inequality always

    involves special assumptions pertaining to the experimental setup, andthe tenacious devils advocate is bound to nd some narrow crack some-where through which a hidden variable or two might slip. In some cases,improvements in the experimental apparatus have sealed a crack shutor have promised to do so (were one to try hard enough, along a coursethat would merely continue a pattern of improvements, say of the ef-ciency of photon detectors); in other cases, a new style of proof ofinequalities has bypassed a putative gap in earlier derivations between a"metaphysical" motivating premise (e.g. separability) and a mathemati-cal condition (e.g. factorizability of certain joint probabilities relative tohypothetical, hidden, "physically complete" states) taken to "precisify"the premise.1 But surely it is in the nature of the beast that there willalways be "loopholes", i.e. some wiggle-room "in principle" for the die-hard hidden-variables advocate, some uncertainty in the case based onexperimental tests of Bell-type inequalities, however carefully and stur-dily it may have been erected. Experimental metaphysics is, after all,experimental, and as Abner Shimony has often emphasized we mustbe fallibilists, recognising the possibility of error but without that at allmuting the voice of reason.To motivate the main focus of this essay, let us recall the case against

    (dynamical) "determinism-in-nature" based on experimental tests ofBell-type inequalities (including the Clauser-Horne-Shimony-Holt [1969]and related inequalities). That case rests on an argument beginning,per reductio,with the assumption of "local determinism", that the ac-tual outomes of (say, for simplicity) spin experiments on each of a pairof spin-1

    2particles prepared in the singlet state, carried out under cir-

    cumstances such that the analyzer-setting (orientation of magnetic eldof Stern-Gerlach devices) and outcome events at opposite wings of thesetup are space-like separated, are determined by physical conditions ona space-like slice restricted to the past light cones, respectively, of theindividual analyzer-setting and outcome events. Those conditions may,of couse, pertain to the local measuring apparatus. But it is required

    1See e.g. Hellman [1992].

    2

  • that physical conditions of the apparatus at the opposite wing not beincluded. Outcomes at each wing are assumed to be physically andstatistically independent of parameter settings at the other wing ("para-meter independence"). This is the "locality" part of "local determinism".(We will assume that this is well-motivated by various arguments basedon limitations on the speed of energy-momentum transfer inherent inthe special and general theories of relativity.) Further, in the ideal case,conservation (of angular momentum) requires that opposite outcomeswill be found in parallel experiments (i.e. same orientation of magneticelds of Stern-Gerlach apparati set up along a common axis at the tworespective wings of the experiment).2 Still these assumptions are notsu cient for a test of "determinism-in-nature". For each (sub-)systemcan be tested only once for a particular orientation of spin and setting ofparameters at the opposite wing. What then is to block a deterministicaccount (theory) of any ensemble of such systems-cum-experiments youlike which just manages to deliver the right actual outcomes for each ex-periment, one-by-one, as it were? By "one-by-one" we dont mean thatsuch a theory simply provides a list of outcomes, and therefore could beexcluded for not being "well-systematized". Rather we mean that thetheory somehow manages to take into account for all we know, perhapsin a unied way only the actual conditions obtaining for the individualsystems involved. The short answer is that such a theory is not "robust"unless it also supports counterfactuals telling us what oucomes wouldhave emerged at a given wing had the parameter settings been dierentat the opposite wing. This, of course, is the same requirement that theEinstein-Podolsky-Rosen [1935] argument invoked in their famous casethat quantummechanics must be "incomplete". In the present setting, itis used to infer that a robust or respectable deterministic theory must de-liver (counterfactual) predictions of outomes at a given wing for variousparameter settings at the opposite wing. (Three directions altogethersu ce, set e.g. 120 apart, for deriving a Bell inequality discriminatinglocal, deterministic hidden variablespredictions from those of quantummechanics.)3 It is assumed that, as in experiments of Aspect, et al., thepreparation of apparatus at an opposite wing being considered is space-like related to the opposite subsystems measurement at the other wingso that it is reasonable to assume that no physical interaction occursbetween these "events" (or space-time regions). Then we can proceed

    2In the case of polarization in photon cascade experiments, (exact) conservationrequires that passage or non-passage through analyzers set at the same orientationat opposite wings be directly strictly correlated.

    3For an elegant presentation and further simplication of an argument due to D.Mermin (itself simplifying one given by E. Wigner), see Kosso [1998], 143-48.

    3

  • as in the standard proofs of Bells theorem, that the theory must de-liver a set of (actually and counterfactually predicted) "values" to thegiven subsystemsspins that respect parameter-independence and thusnecessarily satisfy Bell-type inequalities in cases in which quantum me-chanically predicted statistics violate them.4

    This reasoning makes it clear that the "metaphysics" in experimentalmetaphysics is mediated by requirements that we are led to impose onputative theories that would transcend quantum mechanics but accountfor the observed statistics. That should not be surprising, however, sincethe metaphysical words (such as "local determinism") must be spelledout carefully if we are to carry out a mathematical argument constrainingpossible explanations of the observations, and of course explanations inphysics, at any rate, typically involve some theory.5

    If the physical world, at least at the quantum level, is really inde-terministic in the ways described by the Bell results just outlined, it isnatural to speak of inidivual outcomes in tests for spin or polarization as"objectively random", in that literally nothing in nature causes any ofthose particular outcomes (as opposed to the opposite value of the rele-vant two-valued observable). If we think of trying to connect this withvarious mathematical denitions of "random sequence" (of digits), wecan imagine generating sequences of outcomes (coded by, say, 0s and 1s,respectively, for the two possible outcomes at each wing, L and R, takenseparately) by repeating "identically prepared" experiments many timesand checking the relevant formal properties exhibited by the outcomesat a given wing. (Clearly, this will be rather easier if the mathematicaldenition of "random sequence", such as that of Kolmogorov-Chaitin,applies to nite sequences!) But we will not expect such sequences to beentirely random or chaotic in an intuitive sense. Rather we expect that,in almost every case, they will exhibit convergence of ratios presentedin the initial segments to probability values given by quantum theory.(For example, in the case o