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arXiv:0910.0169

Quantum Behavior in Machian Model

Merab Gogberashvili

Published Oct 1, 2009• 1 citation
arXivHTMLPDF

Abstract

The essentials of quantum theory, the Schrödinger equation and the Planck constant, are derived using classical statistical mechanics within the non-local Machan model. The appearance of complex wave function is connected with the necessity of quantization condition when one neglects the existence of the preferred frame and assumes Galilean symmetry of the model.

DOI
10.48550/arxiv.0910.0169

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The essentials of quantum theory, the Schrödinger equation and the Planck constant, are derived using classical statistical mechanics within the non-local Machan model. The appearance of complex wave function is connected with the necessity of quantization condition when one neglects the existence of the preferred frame and assumes Galilean symmetry of the model.
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.
A version of quantum theory is derived from a set of plausible assumptions related to the following general setting: For a given system there is a set of experiments that can be performed, and for each such experiment an ordinary statistical model is defined. The parameters of the single experiments are functions of a hyperparameter, which defines the state of the system. There is a symmetry group acting on the hyperparameters, and for the induced action on the parameters of the single experiment a simple consistency property is assumed, called permissibility of the parametric function. The other assumptions needed are rather weak. The derivation relies partly on quantum logic, partly on a group representation of the hyperparameter group, where the invariant spaces are shown to be in 1-1 correspondence with the equivalence classes of permissible parametric functions. Planck's constant only plays a role connected to generators of unitary group representations.
The nature of quantum description is clarified. It is shown that complex-valued probability amplitudes are admissible within classical Hamiltonian mechanics. According to standard probability theory, such a description is always possible. The case of a spherical phase space is considered. It is shown that, in such a classical theory, there appears a universal constant that has dimensions of action ( h ), as well as Fock space and all attributes of quantum mechanics. Excitations of a chain of such systems are described by the equations of quantum mechanics with a correct normalization condition. It is shown that an answer to the question of what a particle and its wave function are is provided by quantum field theory (these are a single-particle field excitation and a function that describes it). Experiments are proposed that would make it possible to solve the problem of the physical nature of the wave function.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
A research program within the scope of theories on "Emergent Quantum Mechanics" is presented, which has gained some momentum in recent years. Via the modeling of a quantum system as a non-equilibrium steady-state maintained by a permanent throughput of energy from the zero-point vacuum, the quantum is considered as an emergent system. We implement a specific "bouncer-walker" model in the context of an assumed sub-quantum statistical physics, in analogy to the results of experiments by Couder and Fort on a classical wave-particle duality. We can thus give an explanation of various quantum mechanical features and results on the basis of a "21st century classical physics", such as the appearance of Planck's constant, the Schrödinger equation, etc. An essential result is given by the proof that averaged particle trajectories' behaviors correspond to a specific type of anomalous diffusion termed "ballistic" diffusion on a sub-quantum level. It is further demonstrated both analytically and with the aid of computer simulations that our model provides explanations for various quantum effects such as double-slit or n-slit interference. We show the averaged trajectories emerging from our model to be identical to Bohmian trajectories, albeit without the need to invoke complex wavefunctions or any other quantum mechanical tool. Finally, the model provides new insights into the origins of entanglement, and, in particular, into the phenomenon of a "systemic" non-locality.