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AIP conference proceedings

Symplectic geometry on the Hilbert phase space and foundations of quantum mechanics

Andrei Yu. Khrennikov

Published Jan 1, 2006• 3 citations
Publisher

Abstract

We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite‐dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite‐dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows.

DOI
10.1063/1.2205816

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We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite‐dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite‐dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows.
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 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 .
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 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 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.
The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.