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arXiv:1703.00309

Collapse of Probability Distributions in Relativistic Spacetime

Hans C. Ohanian

Published Mar 1, 2017• 0 citations
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Abstract

The collapse of a spatial probability distribution is triggered by a measurement at a given spacetime point. It is customarily assumed that this collapse occurs along an equal-time hypersurface, say, t = 0. However, such a naïve instantaneous collapse process is inconsistent with relativity, because the equal-time hypersurfaces of different inertial reference frames are different. The attempts at implementation of instantaneous collapse in several different reference frames then lead to violations of probability conservation and violations of the scalar character of the probability contained in given volume elements. This problem affects not only the Copenhagen interpretation of quantum mechanics, but also other interpretations in which it is still necessary to specify what changes in probabilities occur when and where in a manner consistent with relativistic spacetime geometry. In the 1980s Schlieder and Hellwig and Kraus proposed that collapse of the probability distribution along the past light cone of the measurement point avoids these difficulties and leads to a Lorentz-invariant collapse scenario. Their proposal received little attention and some negative criticisms. In this paper I argue that the proposed past-light cone collapse is not only reasonable, but is compelled by Lorentz invariance of probability conservation, and is equally valid for the spatial probability distributions in quantum mechanics and for those in a game of chance, for instance, the probability distribution for a game with playing cards scattered over some spatial region. I examine the objections that have been made to the past-light-cone collapse scenario and show that these objections are not valid. Finally, I propose two possible interferometer experiments that can serve as direct tests of past-light-cone collapse, one with an atom interferometer, and the other with a light interferometer.

DOI
10.48550/arxiv.1703.00309

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The collapse of a spatial probability distribution is triggered by a measurement at a given spacetime point. It is customarily assumed that this collapse occurs along an equal-time hypersurface, say, t = 0. However, such a naïve instantaneous collapse process is inconsistent with relativity, because the equal-time hypersurfaces of different inertial reference frames are different. The attempts at implementation of instantaneous collapse in several different reference frames then lead to violations of probability conservation and violations of the scalar character of the probability contained in given volume elements. This problem affects not only the Copenhagen interpretation of quantum mechanics, but also other interpretations in which it is still necessary to specify what changes in probabilities occur when and where in a manner consistent with relativistic spacetime geometry. In the 1980s Schlieder and Hellwig and Kraus proposed that collapse of the probability distribution along the past light cone of the measurement point avoids these difficulties and leads to a Lorentz-invariant collapse scenario. Their proposal received little attention and some negative criticisms. In this paper I argue that the proposed past-light cone collapse is not only reasonable, but is compelled by Lorentz invariance of probability conservation, and is equally valid for the spatial probability distributions in quantum mechanics and for those in a game of chance, for instance, the probability distribution for a game with playing cards scattered over some spatial region. I examine the objections that have been made to the past-light-cone collapse scenario and show that these objections are not valid. Finally, I propose two possible interferometer experiments that can serve as direct tests of past-light-cone collapse, one with an atom interferometer, and the other with a light interferometer.
A thought experiment is presented that provides a hint as to how quantum wave functions can collapse in a relativistically consistent way. The thought experiment consists of a central source of photon pulses midway between two detectors, which is examined relativistically from both the lab frame and a frame moving with respect to the source and detectors. The total number of pulses between the two detectors is the only invariant and this leads to the idea that collapse might occur along special spacetime intervals, which we call symmetric spacetime intervals. In the current formalism of quantum mechanics, collapse is envisioned to occur “instantaneously”, in clear violation of relativity. It is shown that what the current formalism actually specifies is that collapse is merely spacelike. However, if quantum collapse occurs along symmetric spacetime intervals, then collapse of the wave function can be made consistent with relativity and all observers can agree upon which of two spacelike-separated causally connected events is the cause and which the effect. Nomenclature x = normalized distance in light seconds (ls) t = normalized time in seconds (s) γ = Lorentz factor s = spacetime interval u = normalized velocity (ls/s) c = speed of light (1.0 ls/s) y = a quantum object z = a quantum object d = separation between pulses (ls) L = separation between detectors (ls) ν = frequency (s-1) n = number of pulses β = slope of the symmetric interval (s/ls) I.
In the current formalism of quantum mechanics, collapse of the wave function is envisioned as occurring “instantaneously”. Since this is in conflict with the fact that “simultaneity” is not a relativistic invariant, quantum mechanics and relativity are in conflict with each another. This conflict is benign as long as faster-than-light (FTL) phenomena are disallowed. This paper explores this conflict and considers the implications if collapse can be specified to occur in a way that is consistent with relativity. In particular, relativistically consistent wave function collapse forces us to reconsider our preconceptions of causality. We are led to consider two cases; timelike and spacelike. For two events that are causally connected by a timelike interval, the earlier-in-time event is the cause in all reference frames, but the spatial ordering is not preserved. For two events that are causally connected by a spacelike interval, the temporal ordering is not preserved but the spatial ordering is, just the reverse. This is philosophically disturbing since for spacelike causes half the reference frames see the cause as proceeding backwards in time. However, in and of itself this is not logically inconsistent. Backwards-in-Time Causality (BTC) seems at first glance to permit temporal paradox, since two spacelike causal connections can be used to create a closed causal loop in spacetime. While temporal paradox is logically inconsistent, BTC can still be allowed if there is also a censor mechanism that prevents temporal paradox. In relativity, FTL and BTC are two sides of the same coin, so the major implication of relativistically consistent wave function collapse is that quantum systems should be able to support both FTL and BTC phenomena despite Eberhard’s “proof.”
Collapse as Category Error: Why Wave Function Collapse Cannot Be a Spacetime Event (v2.0) This paper argues that the demand for wave function collapse to be a spacetime-localised physical event constitutes a category error in relativistic quantum mechanics. In scenarios involving spacelike-separated measurements on entangled systems, different inertial frames assign different temporal orderings to measurement events. Treating collapse as a physical event with a definite time and place therefore requires a preferred foliation of spacetime, in violation of Lorentz covariance. The paper shows that all empirically accessible quantities—local measurement outcomes and their joint probability distributions—are Lorentz-invariant and can be derived without invoking any spacetime collapse event. “Collapse” is instead most coherently understood as a relational, non-dynamical re-description of correlation structure conditioned on local measurement context. The contribution is strictly one of conceptual clarification and structural constraint analysis. The paper does not propose a new interpretation of quantum mechanics, introduce dynamical collapse models, or claim to resolve the measurement problem. Version 2.0 extends the analysis with: (i) a structured evaluation matrix (HATI-QM) for comparing interpretational frameworks against Lorentz covariance and epistemic transparency; (ii) an explicit falsifiability architecture (“semantic kill-switch”); and (iii) a formal mapping to the Ecological Homeostasis framework for distributed adaptive systems, identifying collapse localisation as an instance of a broader “pathological localisation” error. The paper is intended for researchers in quantum foundations, philosophy of physics, and information-theoretic approaches to quantum theory.
I propose to resolve the controversy over the speed of collapse of quantum-mechanical wavefunctions by means of an experimental test with a modified symmetric Mach-Zehnder atom interferometer, with non-intersecting, parallel, widely separated final beams. According to the conventional collapse scenario, the coherent twin-peak atomic wavefunction in the beams of the interferometer suffers an instantaneous collapse, at infinite speed, when the atom is captured by one of the two detectors at the ends of the beams, but it remains coherent until that instant. In contrast, according to the Hellwig-Kraus relativistic collapse scenario, the wavefunction collapses at the speed of light, backward in time along the past light cone of each detector. This leads to a premature collapse, or pre-collapse, which for a beam-to-beam separation of 3 m extends over a time span of 10 ns before arrival at the detectors. Within this time span the paired wavepackets in the two beams will be incoherent. The difference between the coherent wavepackets of the conventional scenario and the incoherent wavepackets of the relativistic scenario can be tested by probing the atomic beams with a transverse laser beam crossing them near the detectors. If the paired atomic wavepackets in the two beams are coherent, the light scattered by the two beams will also be coherent and generate a standing light wave in the space between the beams, with detectable interference fringes. If the paired wavepackets are incoherent, no such interference fringes will be generated, and the distribution of the scattered light will be that of two independent dipoles. The design parameters for this test seem to lie within reach of the techniques of atom interferometry.
The changes that quantum states undergo during measurement are both probabilistic and nonlocal. These two characteristics complement one another to insure compatibility with relativity and maintain conservation laws. The probabilistic nature of nonlocal effects prevents the superluminal transmission of information, while nonlocal entanglement relations provide a means to enforce conservation laws in a probabilistic theory. In order to explain measurement-induced changes in terms of fundamental physical processes it is important to take these two key characteristics into account. One way to do this is to modify the Schroedinger equation by adding stochastic, nonlinear terms. A number of such proposals have been made over the past few decades. A recently proposed equation based on the assumption that wave function collapse is induced by a sequence of correlating interactions of the kind that constitute measurements has been shown to maintain strict adherence to conservation laws in individual instances, and has also eliminated the need to introduce any new, ad hoc physical constants. In this work it is shone that the proposed stochastic modification to the Schroedinger equation is also Lorentz invariant, even though it is formulated in a preferred reference frame. It is further argued that the additional spacetime structure that the proposed modification requires provides a way to implement the assumption that spacelike-separated operators (and measurements) commute, and that this assumption of local commutativity should be regarded as a third postulate of relativity.
In this paper we introduce the three main notions of probability used by physicists and discuss how these are to be used when invoking spacelike separated observers in a relativistic format. We discuss a standard EPRB experiment and concentrate upon problems of the interpretation of probabilities. We promote a particularly conservative interpretation of this experiment (which need not invoke an objective notion of collapse) where probabilities are, tentatively, passively Lorentz invariant. We also argue that the Heisenberg picture is preferable in relativistic situations due to a conflict between the Schrodinger picture and passive Lorentz transformations of probabilities. Throughout most of this paper we discuss the relative frequency interpretation of probability as this is most commonly used. We also introduce the logically necessary notion of `prior-frequency' in discussing whether the choice by an observer can have any causal effect upon the measurement results of another. We also critically examine the foundational use of relative frequency in no-signalling theorems. We argue that SQT and SR are probabilistically compatible, although we do not discuss whether they are compatible on the level of individual events.