Path integrals for the relativistic particle: Some conceptual and pedagogical comments

TL;DR

This paper discusses a lattice regularization approach for the relativistic particle's path integral, clarifying conceptual issues and exploring pedagogical implications, while highlighting open questions in the field.

Contribution

It provides a detailed clarification of the lattice regularization method for the relativistic particle's path integral and discusses its conceptual and pedagogical aspects.

Findings

Clarification of lattice regularization for relativistic path integrals

Interpretation of results via the Jacobi action principle

Identification of open questions in the approach

Abstract

In my textbook on Quantum Field Theory \cite{tpqft} and in a recent paper \cite{tpejc2018}, I advocated a lattice regularization procedure for defining the path integral for the relativistic particle, using the non-quadratic action containing a square root. I also provided an interpretation of this result in terms of the Jacobi action principle. This note clarifies several conceptual and pedagogical issues related to this approach and highlights some interesting open questions which this result leads to.