Differential Equations and Mathematical Physics

Lagrange's representation of the quantum evolution of matter fields

A. Yu. Samarin, A. M. Shterenberg

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: It is shown that a quantum path integral can be represented as a functional of the unique path that satisfies the principle of least action. The concept of path will be used, which implies the parametric dependence of the coordinates of a point on time $x(t)$, $y(t)$, $z(t)$. On this basis, the material fields, which are identified with a quantum particle, are represented as a continuous set of individual particles, the mechanical motion of which determines the spatial fields of the corresponding physical quantities. The wave function of a stationary state is the complex density of matter field individual particles. The modulus of complex density sets the density of matter normalized in one way or another at a given point in space, and the phase factor determines the result of the superposition of material fields in it. This made it possible to transform the integral equation of quantum evolution to the Lagrange's representation. By using the description of a quantum harmonic oscillator as an example, this approach is verified. EPR-type experiment is described in detail, and the possibility of the faster-then light communication is proved, as well as the possible rules of thumb of this communication are proposed.

Keywords: path integrals, matter field, nonlocality, quantum evolution in physical space, Lagrange's and Euler's representations, quantum dynamics of a closed system.

UDC: 517.958:530.145:512

MSC: 81S40, 58D30

Received: September 18, 2022
Revised: December 25, 2022
Accepted: January 24, 2023
First online: March 19, 2023

DOI: 10.14498/vsgtu1953

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