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QuantumMechanics as Approximation of Statistical Mechanics of Classical Fields

Andrei Yu. Khrennikov

Published Mar 19, 2014• 0 citations
Publisher

Abstract

In Section 3.1, we showed that QM can be represented as a classical field model in which physical quantities are given by quadratic functionals of fields. In this chapter we shall show that in complete accordance with Einstein’s views QM is an approximation of CSM of fields, i.e., classical statistical mechanics with the infinite- dimensional phase space H = L2(R3;R) × L2(R3;R), where L2(R3;R) is the space of real-valued square-integrable functions. (In the previous considerations the symbol L2(R3) was used to denote the space of complex-valued square-integrable functions.) QM can be represented as the quadratic Taylor approximation of CSM of fields. By expanding classical quantities, functionals of the prequantum field, φ → f (φ), into the Taylor series up to the second term, we obtain quantum quantities. By averaging these expansions we establish a coupling between classical average (so to say, prequantum) and quantum average. The latter appears as the main term in the expansion of the classical phase space average with respect to a small parameter α, the dispersion of the prequantum fluctuations, α = σ 2(μ). Here μ is the probability distribution of a prequantum random field. In the limit α → 0 the classical prequantum and quantum averages coincide. Thus QM can be interpreted as the limit (as α → 0) of the theory of classical random fields, cf. Section 3.11 for an experimentwhichmay show a deviation from the prediction of QM, namely, violation of Born’s rule.

DOI
10.1201/b15540-8

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In Section 3.1, we showed that QM can be represented as a classical field model in which physical quantities are given by quadratic functionals of fields. In this chapter we shall show that in complete accordance with Einstein’s views QM is an approximation of CSM of fields, i.e., classical statistical mechanics with the infinite- dimensional phase space H = L2(R3;R) × L2(R3;R), where L2(R3;R) is the space of real-valued square-integrable functions. (In the previous considerations the symbol L2(R3) was used to denote the space of complex-valued square-integrable functions.) QM can be represented as the quadratic Taylor approximation of CSM of fields. By expanding classical quantities, functionals of the prequantum field, φ → f (φ), into the Taylor series up to the second term, we obtain quantum quantities. By averaging these expansions we establish a coupling between classical average (so to say, prequantum) and quantum average. The latter appears as the main term in the expansion of the classical phase space average with respect to a small parameter α, the dispersion of the prequantum fluctuations, α = σ 2(μ). Here μ is the probability distribution of a prequantum random field. In the limit α → 0 the classical prequantum and quantum averages coincide. Thus QM can be interpreted as the limit (as α → 0) of the theory of classical random fields, cf. Section 3.11 for an experimentwhichmay show a deviation from the prediction of QM, namely, violation of Born’s rule.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω = H × H , where H is the real Hilbert space. Statistical states are given by Gaussian measures on Ω having zero mean value and dispersion of very small magnitude α (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 → 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) of variables f : Ω → R with respect to the small parameter √α The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space 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. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.
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 (in contrast to rather common opinion) QM might be not a complete theory. We present a prequantum model with the infinite dimensional phase space such that QM can be considered as an algorithm for approximation of averages with respect to prequantum fluctuations. Such an approximation is based on the asymptotic expansion of classical statistical averages with respect to a small parameter &kgr;. Therefore statistical predictions of QM are only approximative and a better precision of measurements would induce deviations of experimental averages from quantum mechanical ones. In this paper we present a natural physical interpretation of &kgr; as the time scaling parameter (between quantum and prequantum times). We analyse consequences of various choices of the prequantum time scale. Possible connections with attophysics and Grand Unification model are discussed.
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.
One of the crucial differences between mathematical models of classical and quantum mechanics (QM) is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems, one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and the quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein dreamed of. QM is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem. We present the basic physical interpretation of prequantum classical statistical field theory in Sec. II. However, this is only the first step toward real physical theory.
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.