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

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.