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arXiv:quant-ph/0505228 · Journal of Physics A Mathematical and General

A pre-quantum classical statistical model with infinite-dimensional phase space

Andrei Yu. Khrennikov

Published Sep 28, 2005• 97 citations
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Abstract

We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R 2 n , but its infinite-dimensional analogue. Here the phase space Ω = H × H , where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f :Ω → R . The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion ≈ h . The quantum statistical model is obtained as the lim h →0 of the classical one. All quantum states including so-called ‘pure states’ (wavefunctions) are simply Gaussian fluctuations of the ‘vacuum field’, ω = 0 ∊ Ω, having dispersions of the Planck magnitude.

DOI
10.1088/0305-4470/38/41/015

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We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R 2 n , but its infinite-dimensional analogue. Here the phase space Ω = H × H , where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f :Ω → R . The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion ≈ h . The quantum statistical model is obtained as the lim h →0 of the classical one. All quantum states including so-called ‘pure states’ (wavefunctions) are simply Gaussian fluctuations of the ‘vacuum field’, ω = 0 ∊ Ω, having dispersions of the Planck magnitude.
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 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 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 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
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.