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arXiv:quant-ph/9905032

Quantum Mechanics as a Classical Field Theory

de la Torre, A. C., Alejandro Daleo

Published May 10, 1999• 0 citations
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Abstract

The formalism of quantum mechanics is presented in a way that its interpretation as a classical field theory is emphasized. Two coupled real fields are defined with given equations of motion. Densities and currents associated to the fields are found with their corresponding conserved quantities. The behavior of these quantities under a galilean transformation suggest the association of the fields with a quantum mechanical free particle. An external potential is introduced in the Lagrange formalism. The description is equivalent to the conventional Schrödinger equation treatment of a particle. We discuss the attempts to build an interpretation of quantum mechanics based on this scheme. The fields become the primary onthology of the theory and the particles appear as emergent properties of the fields. These interpretations face serious problems for systems with many degrees of freedom.

DOI
10.48550/arxiv.quant-ph/9905032

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The formalism of quantum mechanics is presented in a way that its interpretation as a classical field theory is emphasized. Two coupled real fields are defined with given equations of motion. Densities and currents associated to the fields are found with their corresponding conserved quantities. The behavior of these quantities under a galilean transformation suggest the association of the fields with a quantum mechanical free particle. An external potential is introduced in the Lagrange formalism. The description is equivalent to the conventional Schrödinger equation treatment of a particle. We discuss the attempts to build an interpretation of quantum mechanics based on this scheme. The fields become the primary onthology of the theory and the particles appear as emergent properties of the fields. These interpretations face serious problems for systems with many degrees of freedom.
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The present paper is based upon equations obtained in an earlier paper by the author devoted to a new formulation of quantum electrodynamics. The equations describe the structure of the electron as well as its motion in external fields, interaction with a measuring apparatus inclusive, in a deterministic manner without any jumps. Quantum mechanics is an approximate theory because its equations follow from the above equations upon neglecting the self-field of the electron itself. Just this leads to paradoxes, seeming contradictions and jumps in quantum mechanics. The quantum mechanical wavefunction has a dual interpretation. In some problems the square of its modulus represents a real distribution of the electronic density while in others the same square determines the probability distribution of coordinates. It is shown why, given the different interpretations of the wavefunction, it satisfies one and the same Dirac or Schrödinger equation. Description of many-electron systems is also considered in the starting approach as well as in quantum mechanics. Neutrinos are discussed in brief.
In this paper, we introduce "the Schrödinger plate." This is an infinite two-dimensional linear micro-polar elastic medium, with out-of-plane degrees of freedom, lying on a linear elastic foundation of a special kind. Any free motion of the plate can be corresponded to a solution of the two-dimensional Schrödinger equation for a single particle in the external potential field VVV. The specific dependence of the potential VVV on the position is taken into account in the properties of the plate elastic foundation. The governing equations of the plate are derived as equations of the two-dimensional constraint Cosserat continuum using the direct approach. The plate dynamics can be described by the classical Germain-Lagrange equation for a plate, but the strain energy is different from the one used in the classical Kirchhoff-Love plate theory. Namely, the Schrödinger plate cannot be imagined as a thin elastic body composed of an isotropic linear material. The main property of the Schrödinger plate is as follows: the mechanical energy propagates in the plate exactly in the same way as the probability density propagates according to the corresponding Schrödinger equation.
Quantum mechanics is a cornerstone of modern physics, yet its fundamental origin and mathematical foundation remain elusive. In this paper, we derive the complete structure of quantum mechanics from a purely classical field theory. By gauging the double cover of the Lorentz symmetry group of space–time, the resulting equations of motion naturally yield all postulates of quantum mechanics, including the emergence of the Planck constant. This approach provides a coherent interpretation and demonstrates that quantum mechanics can be viewed as a mathematically consistent classical field theory.
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a nonrelativistic quantum-mechanical description of the clock. From the clock-dependent Schr\"odinger equation (as an analog of the time-dependent Schr\"odinger equation) we derive a continuity equation, where, instead of a time derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.