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arXiv:quant-ph/0511074

Quantum mechanics as an asymptotic projection of statistical mechanics of classical fields: derivation of Schrödinger's, Heisenberg's and von Neumann's equations

Andrei Yu. Khrennikov

Published Nov 8, 2005• 1 citation
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Abstract

We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ 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 $f: Ω\to {\bf R}$ (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 $ψ\_{\rm{vacuum}}\equiv 0)$ of variables $f: Ω\to {\bf R}$ with respect to the small parameter $\sqrtα.$ 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.

DOI
10.48550/arxiv.quant-ph/0511074

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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 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 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 the Planck magnitude -- fluctuations of the ``vacuum field.'' Physical variables (e.g., energy) are given by maps f:Ω→Rf: Ω\to {\bf R}f:Ω→R (functions of classical fields). The crucial point is that statistical states and variables are symplectically invariant. The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Teylor expansion (up to the terms of the second order at the vacuum field point ω=0)ω=0)ω=0) of variables f:Ω→Rf: Ω\to {\bf R}f:Ω→R with respect to the small parameter κ=h.κ= \sqrt{h}.κ=h​. A Gaussian symplectically invariant measure (statistical state) is represented by its covariation operator (von Neumann statistical operator). A symplectically invariant smooth function (variable) is represented by its second derivative at the vacuum field point ω=0.ω=0.ω=0. From the statistical viewpoint QM is a statistical approximation of the prequantum classical statistical field theory (PCSFT). Such an approximation is obtained through neglecting by statistical fluctuations of the magnitude o(h),h→0,o(h), h\to 0,o(h),h→0, in averages of physical variables. Equations of Schrödinger, Heisenberg and von Neumann are images of dynamics on ΩΩΩ with a symplectically invariant Hamilton function.
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 study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of “dequantization” of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of “prequantum classical statistical field theory.”
We study the relation between the mathematical structures of statistical mechanics on an infinite-dimensional phase space (denoted by ) and quantum mechanics. It is shown that quantum averages (given by the von Neumann trace formula) can be obtained as the main term of the asymptotic expansion of Gaussian functional integrals with respect to a small parameter . Here is the dispersion of the Gaussian measure. The symplectic structure on the infinite-dimensional phase space plays a crucial role in our considerations. In particular, the Gaussian measures that induce quantum averages must be consistent with the symplectic structure. The equations of Schrodinger, Heisenberg and von Neumann are images of the Hamiltonian dynamics on .