Formal similarity between mathematical structures of electrodynamics and quantum mechanics

TL;DR

This paper reveals a formal analogy between the mathematical structures of electrodynamics and quantum mechanics by reformulating the Schrödinger equation using a real scalar field, highlighting deep structural similarities.

Contribution

It introduces a Lagrangian formulation of the Schrödinger equation with a real scalar field, establishing an analogy to electrodynamics and linking probability conservation to energy conservation.

Findings

The Schrödinger equation can be reformulated as a real scalar field theory.

The mathematical structure of quantum mechanics parallels that of electrodynamics.

Probability conservation corresponds to energy conservation in the scalar field.

Abstract

Electromagnetic phenomena can be described by Maxwell equations written for the vectors of electric and magnetic field. Equivalently, electrodynamics can be reformulated in terms of an electromagnetic vector potential. We demonstrate that the Schr\"odinger equation admits an analogous treatment. We present a Lagrangian theory of a real scalar field $\phi$ whose equation of motion turns out to be equivalent to the Schr\"odinger equation with time independent potential. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics. The field $\phi$ is in the same relation to the real and imaginary part of a wave function as the vector potential is in respect to electric and magnetic fields. Preservation of quantum-mechanics probability is just an energy conservation law of the field $\phi$.