This article is the third of a trilogy of articles on the nature of probability in quantum mechanics. The first article [1] began by noting that superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into the position probability density of an atomic particle, which happens nowhere else in probability theory. It went on to also note that there is an unexplained coincidence in quantum mechanics in that the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables and went on to examine whether there could be an archetypical variable, in the Platonic sense of true form, behind quantum probability that would reconcile quantum probability with classic probability. This examination found such a variable, which can encompass both local and nonlocal quantum events. The second article [2] provided evidence that quantum probability has such a stochastic nature. This evidence was based on the number of electrons that need to be sent through a two-slit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process. This final article completes the trilogy by considering how a key aspect of experimental design would be affected by the increased variability that would be present if quantum probability is itself stochastic in the manner suggested in the previous two articles.
This article is the third of a trilogy of articles on the nature of probability in quantum mechanics. The first article [1] began by noting that superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into the position probability density of an atomic particle, which happens nowhere else in probability theory. It went on to also note that there is an unexplained coincidence in quantum mechanics in that the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables and went on to examine whether there could be an archetypical variable, in the Platonic sense of true form, behind quantum probability that would reconcile quantum probability with classic probability. This examination found such a variable, which can encompass both local and nonlocal quantum events. The second article [2] provided evidence that quantum probability has such a stochastic nature. This evidence was based on the number of electrons that need to be sent through a two-slit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process. This final article completes the trilogy by considering how a key aspect of experimental design would be affected by the increased variability that would be present if quantum probability is itself stochastic in the manner suggested in the previous two articles.
This report covers work since a previous article on quantum probability. It provides evidence that quantum probability has a stochastic nature. This evidence is based on the number of electrons that need to be sent through a twoslit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process.
Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it is the stochastic archetype of quantum probability.
INTRODUCTION The purpose of this article is to bring out in as conceptually clear terms as possible what seems to be a major incompleteness in the probability theory of particles offered by classical quantum mechanics. The exact nature of this incompleteness is illustrated by consideration of some simple quantum-mechanical examples. In addition, these examples are contrasted with the fundamental assumptions of Brownian motion in classical physics on the one hand, and with a controversy of a decade ago in mathematical psychology. (The psychological example is described in detail in the appendix). Our central claim is that classical quantum mechanics is radically incomplete in its probabilistic account of the motion of particles. In the last part of the article we derive the time-dependent joint distribution of position and momentum of the linear harmonic oscillator, and show how the apparently physically paradoxical statistical independence of position and momentum has a natural explanation. The explanation is given within the framework of the non-quantum-mechanical stochastic theory we construct for such oscillators. Before entering into any technical details, we shall attempt to spell out in intuitive terms our incompleteness claim. To begin with, there are two senses of completeness that we want to distinguish. One sense of completeness is that demanded by any hidden variable theories of quantum mechanics. States should be found, or be theoretically characterizable at least, that lead to precise values for all observables; that is, no observable is to have a variance greater than zero.
The basic postulates The statistical interpretation of the wavefunction is in accord with experimental facts. An interference pattern on a screen is built up by a series of apparently random events, and the wavefunction correctly predicts where the particle is most likely to land over an ensemble of trials (cf. the frontispiece ). Yet the interpretation of the wavefunction which ascribes to it a purely statistical significance is not forced on us by the experimental results – the latter show only that it is consistent to regard the wavefunction as containing information on probability; they do not exclude the possibility that it may possess other properties. On the contrary, one may take the view that the characteristic distribution of spots on a screen which build up an interference pattern is evidence that the wavefunction indeed has a more potent physical role than a mere repository of information on probabilities, for how are the particles guided so that statistically they fall into such a pattern? Such a question is naturally ruled out by the purely probabilistic interpretation. But the latter is appropriate only if we wish to reduce physics to a kind of algorithm which is efficient at correlating the statistical results of experiments. If we wish to do more, and attempt to understand the experimental results as the outcome of a causally connected series of individual processes, then we are free to enquire as to the possible further significance of the wavefunction (beyond its probabilistic aspect), and to introduce other concepts in addition to the wavefunction.
The stochastic nature of quantum mechanics is more naturally reflected in a bilinear two-process representation of density matrices rather than in squared wave functions. This proposition comes with a remarkable change of the entanglement mechanism: entanglement effects do not originate from superpositions of wave functions, but result from the bilinear structure of density matrices. Quantum interference appears as a multiplicative phenomenon rather than an additive superposition mechanism. We propose two general requirements such that the bilinear representation of density matrices is given in terms of two uniquely defined, identically distributed, Markovian stochastic jump processes. These general ideas are illustrated for the Einstein-Podolsky-Rosen and double-slit experiments. The expression of the stochastic nature of quantum mechanics in terms of random variables rather than their probability distributions facilitates an ontological viewpoint and leads us to a bra-ket interpretation of quantum mechanics.
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.