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HNPS Advances in Nuclear Physics

Liouville's theorem and quantum mechanics - time quantization and reality

C. Syros, George S. Ioannidis, George Raptis

Published Feb 11, 2020• 0 citations
Publisher

Abstract

The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.

DOI
10.12681/hnps.2798

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The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.
The essentials of quantum mechanics are derived from Liouville's theorem in statistical mechanics. An elementary solution, g, of Liouville's equation helps to construct a differentiable N-particle distribution function (DF), F(g), satisfying the same equation. Reality and additivity of F(g): (i) quantize the time variable; (ii) quantize the energy variable; (iii) quantize the Maxwell–Boltzmann distribution; (iv) make F(g) observable through time-elimination; (v) produce the Planck constant; (vi) yield the black-body radiation spectrum; (vii) support chronotopology introduced axiomatically; (viii) the Schrödinger and the Klein–Gordon equations follow. Hence, quantum theory appears as a corollary of Liouville's theorem. An unknown connection is found allowing the better understanding of space-times and of these theories.
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A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.
We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant ααα with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t)Ψ(x,t)Ψ(x,t) can be expanded in a complete set of functions Φn(x)Φ_n(x)Φn​(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t)W(x,p,t)W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant ααα appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ\hbarℏ.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0\hbar\mapsto0ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.