Path integral of the relativistic point particle in Minkowski space

TL;DR

This paper investigates the symmetries of the relativistic point particle action, identifies a hidden local symmetry, and constructs various relativistic propagators through a topological path integral approach, connecting to the Feynman checkerboard model.

Contribution

It reveals a hidden local symmetry in the relativistic particle action and constructs multiple propagators via a topological path integral method.

Findings

Identified a hidden local symmetry in the relativistic particle action.

Constructed three types of relativistic propagators from different Minkowski sectors.

Connected the path integral approach to the Feynman checkerboard model.

Abstract

In this article, we analyze the fundamental global and local symmetries involved in the action for the free relativistic point particle. Moreover, we identify a hidden local symmetry, whose explicit consideration and factorization utilizing of a Fujikawa prescription, leads to the construction of relativistic propagators that satisfy the Chapman-Kolmogorov identity. By means of a detailed topological analysis, we find three different relativistic propagators (orthochronous, space-like, and Feynman) which are obtained from the exclusive integration of paths within different sectors in Minkowski space. Finally, the connection of this approach to the Feynman checkerboard construction is explored.