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arXiv:hep-th/0604163 · Il Nuovo Cimento B

On the problem of hidden variables for quantum field theory

Andrei Yu. Khrennikov

Published Apr 23, 2006• 11 citations
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Abstract

We show that QFT (as well as QM) is not a complete physical theory. We constructed a classical statistical model inducing quantum field averages. The phase space consists of square integrable functions, $f(ϕ),$ of the classical bosonic field, $ϕ(x).$ We call our model prequantum classical statistical field-functional theory -- PCSFFT. The correspondence between classical averages given by PCSFFT and quantum field averages given by QFT is asymptotic. The QFT-average gives the main term in the expansion of the PCSFFT-average with respect to the small parameter $α$ -- dispersion of fluctuations of "vacuum field functionals.'' The Scrödinger equation of QFT is obtained as the Hamilton equation for functionals, $F(f),$ of classical field functions, $f(ϕ).$ The main experimental prediction of PCSFFT is that QFT gives only approximative statistical predictions that might be violated in future experiments.

DOI
10.48550/arxiv.hep-th/0604163

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We show that QFT (as well as QM) is not a complete physical theory. We constructed a classical statistical model inducing quantum field averages. The phase space consists of square integrable functions, f(ϕ),f(ϕ),f(ϕ), of the classical bosonic field, ϕ(x).ϕ(x).ϕ(x). We call our model prequantum classical statistical field-functional theory -- PCSFFT. The correspondence between classical averages given by PCSFFT and quantum field averages given by QFT is asymptotic. The QFT-average gives the main term in the expansion of the PCSFFT-average with respect to the small parameter ααα -- dispersion of fluctuations of "vacuum field functionals.'' The Scrödinger equation of QFT is obtained as the Hamilton equation for functionals, F(f),F(f),F(f), of classical field functions, f(ϕ).f(ϕ).f(ϕ). The main experimental prediction of PCSFFT is that QFT gives only approximative statistical predictions that might be violated in future experiments.
We show that the probabilistic formalism of QM can be obtained as a special projection of classical statistical mechanics for systems with an infinite number of degrees of freedom. Such systems can be interpreted as classical fields. Thus in our approach QM is a projection of (prequantum) classical statistical field theory (PCSFT). This projection is based on the Taylor expansion of classical physical variables – maps f: Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and negligibly small dispersion. On one hand, the creation of such a prequantum model strongly supports attempts (first of all by Schrödinger and Einstein) to create purely field model of QM. On the other hand, it gives the possibility to go beyond QM. The main experimental prediction is that averages calculated in the mathematical formalism of QM (von Neumann’s trace formula) are only approximative averages. If predictions of PCSFT are correct then it would be possible to find deviations of experimental averages from quantum ones. 1
We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high — the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as the second order derivatives of functionals of classical fields: f(ψ) → Aquant = f″(0). Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion α which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach can be called Prequantum Classical Statistical Field Theory — PCSFT.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω=H×H,Ω= H\times H,Ω=H×H, where HHH is the real Hilbert space. Statistical states are given by Gaussian measures on ΩΩΩ having zero mean value and dispersion of very small magnitude ααα (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps f:Ω→Rf: Ω\to {\bf R}f:Ω→R (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point ψ_vacuum≡0)ψ\_{\rm{vacuum}}\equiv 0)ψ_vacuum≡0) of variables f:Ω→Rf: Ω\to {\bf R}f:Ω→R with respect to the small parameter α.\sqrtα.α​. The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space Ω.Ω.Ω. A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on Ω.Ω.Ω. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.
We show that the mathematical formalism of quantum mechanics can be interpreted as a method for approximation of classical (measure-theoretic) averages of functions f : L2(R3) → R. These are classical physical variables in our model with hidden variables - Prequantum Classical Statistical Field Theory (PCSFT). In this paper we provide a simple stochastic picture of such a quantum approximation procedure. In the probabilistic terms this is nothing else than the approximative method for computation of averages for functions of random variables. Since in PCSFT the space of hidden variables is L2(R3), the role of a classical random variable is played by a random field. In PCSFT we consider Gaussian random fields representing random fluctuations at the prequantum length scale. Quantum mechanical expression for the average (given by the von Neumann trace formula) is obtained through moving from the prequantum length scale to the quantum one (the scale at that we are able to perform measurements).
We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. ``A quantum system in a stationary state ψψψ'' in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of {\it stability of hydrogen atom.