Path Integrals over Velocities in Quantum Mechanics

TL;DR

This paper explores path integrals over velocities in quantum mechanics, providing a unified bosonic framework for propagators in both nonrelativistic and relativistic cases, enabling efficient calculations in external fields.

Contribution

It introduces a universal technique for representing propagators via velocity path integrals without boundary restrictions, applicable to various quantum fields and external conditions.

Findings

Derived explicit forms of propagators in external electromagnetic fields.

Established rules for Gaussian integrals over velocities.

Demonstrated effective calculation of Klein-Gordon propagator in complex fields.

Abstract

Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field.