Magnetic Relativistic Schr\"odinger Operators and \\Imaginary-time Path Integrals

TL;DR

This paper investigates three magnetic relativistic Schr"odinger operators, analyzing their differences, gauge covariance, and path integral representations for their heat equations, contributing to the understanding of their mathematical and physical properties.

Contribution

It introduces and compares three quantizations of relativistic Schr"odinger operators with magnetic fields and reviews their path integral representations for heat equations.

Findings

Operators coincide in constant magnetic fields

Operators differ in general magnetic fields

Path integral representations are established for heat equations

Abstract

Three magnetic relativistic Schr\"odinger operators corresponding to the classical relativistic Hamiltonian symbol with magnetic vector and electric scalar potentials are considered, dependent on how to quantize the kinetic energy term $\sqrt{(\xi-A(x))^2 +m^2}$. We discuss their difference in general and their coincidence in the case of constant magnetic fields, and also study whether they are covariant under gauge transformation. Then results are reviewed on path integral representations for their respective imaginary-time relativistic Schr\"odinger equations, i.e. heat equations, by means of the probability path space measure related to the L\'evy process concerned.