On Three Imaginary-time Path Integral Formulas with Magnetic Fields in Relativistic Quantum Mechanics

TL;DR

This paper develops three path integral formulas for magnetic relativistic Schr"odinger operators, providing new representations of their heat equations using Levy processes and time-sliced approximations.

Contribution

It introduces novel path integral representations for three magnetic relativistic Schr"odinger operators, combining Levy process measures and Chernoff's theorem methods.

Findings

Path integral formulas using Levy processes.

Path integral formulas via time-sliced approximation.

Applicable to magnetic relativistic Schr"odinger operators.

Abstract

Three magnetic relativistic Schr\"odinger operators are considered, corresponding to the classical relativistic Hamiltonian symbol with both magnetic vector and electric scalar potentials. Path integral representations for the solutions of their respective imaginary-time relativistic Schr\"odinger equations, i.e. heat equations are given in two ways. The one is by means of the probability path space measure coming from the L\'evy process concerned, and the other is through time-sliced approximation with Chernoff's theorem.