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arXiv:2410.18324

Hidden variable theory for non-relativistic QED: the critical role of selection rules

Ishikawa, H.

Published Oct 23, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum theory, despite its remarkable success, struggles to represent certain experimental data, particularly those involving integer functions and deterministic relations between quantum jumps. We address this limitation by proposing a hidden variable theory compatible with non-relativistic quantum electrodynamics (QED). Our approach introduces logical variables to describe propositions about the occupation of stationary states, based on three key assumptions: (i) the probabilities of propositions are predictable by quantum theory, (ii) the truth values of propositions conform to Boolean logic, and (iii) propositions satisfy a novel selection rule: ${\rm{Tr}}[\hat{K}(t)\hatρ\hat{K}^{\dagger}(t)]={\rm{Tr}}[({\mathcal{P}}\hat{K}(t)) \hatρ({\mathcal{P}}\hat{K}(t))^{\dagger}]$. Here, $\hat{K}(t)$ is the product of projection operators for the given propositions, and ${\mathcal{P}}\hat{K}(t)$ denotes $\hat{K}(t)$ rearranged arbitrarily. This selection rule extends the consistency condition in the Consistent Histories approach, relaxing constraints imposed by no-go theorems on logical variable representation, joint probability existence, and measurement contextuality. Consequently, our theory successfully describes the essential properties of individual trials, including the discreteness of quantum jumps, the continuity of classical trajectories, and deterministic relations between entangled subsystems. It achieves this without conflicting with quantum equations of motion, canonical commutation relations, or the operator ordering in quantum coherence functions. The direct description of individual trials made possible by this approach sheds new light on the discrete nature of quantum phenomena and the essence of reality.

DOI
10.48550/arxiv.2410.18324

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Quantum theory, despite its remarkable success, struggles to represent certain experimental data, particularly those involving integer functions and deterministic relations between quantum jumps. We address this limitation by proposing a hidden variable theory compatible with non-relativistic quantum electrodynamics (QED). Our approach introduces logical variables to describe propositions about the occupation of stationary states, based on three key assumptions: (i) the probabilities of propositions are predictable by quantum theory, (ii) the truth values of propositions conform to Boolean logic, and (iii) propositions satisfy a novel selection rule: Tr[K^(t)ρ^K^†(t)]=Tr[(PK^(t))ρ^(PK^(t))†]{\rm{Tr}}[\hat{K}(t)\hatρ\hat{K}^{\dagger}(t)]={\rm{Tr}}[({\mathcal{P}}\hat{K}(t)) \hatρ({\mathcal{P}}\hat{K}(t))^{\dagger}]Tr[K^(t)ρ^​K^†(t)]=Tr[(PK^(t))ρ^​(PK^(t))†]. Here, K^(t)\hat{K}(t)K^(t) is the product of projection operators for the given propositions, and PK^(t){\mathcal{P}}\hat{K}(t)PK^(t) denotes K^(t)\hat{K}(t)K^(t) rearranged arbitrarily. This selection rule extends the consistency condition in the Consistent Histories approach, relaxing constraints imposed by no-go theorems on logical variable representation, joint probability existence, and measurement contextuality. Consequently, our theory successfully describes the essential properties of individual trials, including the discreteness of quantum jumps, the continuity of classical trajectories, and deterministic relations between entangled subsystems. It achieves this without conflicting with quantum equations of motion, canonical commutation relations, or the operator ordering in quantum coherence functions. The direct description of individual trials made possible by this approach sheds new light on the discrete nature of quantum phenomena and the essence of reality.
Quantum trajectory theories have not fully reconciled discrete quantum jumps with continuous unitary evolution. We address this challenge by developing a hidden variable formulation that reveals hidden correlations in individual trials. We represent jump sequences as logical propositions on energy eigenstate occupations, where logical variables describe their truth values and chain operators describe their probabilities. To circumvent established no-go theorems, we introduce a proposition selection rule that requires well-defined joint probabilities for a given density operator. This rule unifies compatibility criteria for quantum and hidden variable theories, eliminating incompatible propositions that lead to value assignment paradoxes. Unlike conventional criteria mandating operator commutativity, it permits propositions that become deterministic for specific initial conditions. This framework systematically captures deterministic relations in individual trials, including the exclusivity of state occupation implied by the observed jumps. This exclusivity, distinct from the Pauli exclusion principle, enables the formalization of a quantum version of Boolean logic as laws for quantum propositions. These assumptions circumvent Bell-type inequalities by prohibiting arithmetic operations between incompatible logical variables, while enabling classical descriptions within the complementarity. The resulting theory reconciles discrete jumps with continuous evolution, providing a direct description of individual trials while preserving standard probability predictions.
Experiments motivated by Bell's theorem have led some physicists to conclude that quantum theory is nonlocal. However, the theoretical basis for such claims is usually taken to be Bell's Theorem, which shows only that if certain predictions of quantum theory are correct, and a strong hidden-variable assumption is valid, then a certain locality condition must fail. This locality condition expresses the idea that what an experimenter freely chooses to measure in one spacetime region can have no effect of any kind in a second region situated spacelike relative to the first. The experimental results conform closely to the predictions of quantum theory in such cases, but the most reasonable conclusion to draw is not that locality fails, but rather that the hidden-variable assumption is false. For this assumption conflicts with the quantum precept that unperformed experiments have no outcomes. The present paper deduces the failure of this locality condition directly from the precepts of quantum theory themselves, in a way that generates no inconsistency or any conflict with the predictions of relativistic quantum field theory.
The nondeterministic character of quantum measurement can, and should, be taken to imply a deeper 'hidden variable' description of a system, which reproduces quantum theory when the unknown values of the variables are marginalised over. Differences in measurements on identically prepared systems then represent differences in the hidden variables of the systems. Not to seek the hidden variables, as the Copenhagen interpretation of quantum mechanics arbitrarily instructs, is to give up all hope of improvement in advance, and is contrary to the purposes of science. Correspondingly, it can never be proven that hidden variables don't exist; the most that can be done is to place hypothetical restrictions on them and prove that the resulting theory cannot be marginalised into quantum theory - implying that the hidden variables violate the restriction. The best known example is Bell's theorem, which rules out all local hidden-variable theories and explains why the variables are so coy: it is not easy to isolate and influence something nonlocal. A promising way forward is to proceed from the Dirac theory of the electron in its natural, geometric language of Clifford algebra. This approach, due to Hestenes, suggests that the free electron is in fact a massless point particle executing zitterbewegung at the speed of light, with radius the electron Compton wavelength. This proposal improves the correspondence of the physics with the formalism and resolves many paradoxes; it also hints that the hidden variables are concerned with those parameters needed to define the electron orbit completely. These ideas all represent a demystification of quantum theory. In the 19th century, microscopes advanced to the point at which a new phenomenon, today called Brownian motion after its chronicler, could be seen. The small particles, newly made visible ('dust' particles), were observed to jiggle around in a manner the scientists could not predict, for no apparent reason. Motion of any individual dust particle could not be predicted; however, statistical details of the motion, such as the average (drift) velocity or the mean square velocity, were reproducible. The scientists of the day, probably without realising it, made an assumption: that there was a reason for the jiggling, and that if they could find this, they could predict the precise motion of individual dust particles. They never said Maybe the dust particles just behave like that and there isn't a why; maybe we can go no further and this is the end of the line. 79
A history of the discovery of quantum mechanics and paradoxes of its interpretation is reconsidered from the modern point of view of quantum stochastics and information. It is argued that in the orthodox quantum mechanics there is no place for quantum phenomenology such as events. The development of quantum measurement theory, initiated by von Neumann, and Bell's conceptual critics of hidden variable theories indicated a possibility for resolution of this crisis. This can be done by divorcing the algebra of the dynamical generators and an extended algebra of the potential (quantum) and the actual (classical) observables. The latter, called beables, form the center of the algebra of all observables, as the only visible (macroscopic) observables must be compatible with any hidden (microscopic) observable. It is shown that within this approach quantum causality can be rehabilitated within an extended quantum mechanics (eventum mechanics) in the form of a superselection rule for compatibility of the consistent histories with the statistically predictable future. The application of this rule in the form of the nondemolition principle leads to the statistical inference of the von Neumann projection postulate, and also to the more general quantum information dynamics for instantaneous events, spontaneous localizations (i.e. quantum jumps), and state diffusions (i.e. continuous trajectories). This gives a dynamical solution, in the form of a Dirac boundary value problem and reduced filtering equations, of the notorious decoherence and measurement problems which was tackled unsuccessfully by many famous mathematicians and physicists starting with von Neumann, Schroedinger and Bohr.
This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as “non-classical”. The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell’s criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics.