The use of exp(iS[x]) in the sum over histories

TL;DR

This paper critically examines the use of the exponential of the action in sum-over-histories formulations, highlighting limitations and clarifying its proper context, especially in relativistic and higher-order systems.

Contribution

It clarifies the conditions under which the sum over histories with $ ext{exp}(iS[x])$ is valid and extends the derivation to relativistic particles, challenging common assumptions.

Findings

The form $ ext{exp}(iS[x])$ is justified only for second-order, non-relativistic systems.

In relativistic cases, path weights differ from $ ext{exp}(iS[x])$, especially for causal and Feynman Green's functions.

The role of inner product and operator ordering critically affects the sum over histories.

Abstract

The use of $\sum \exp(iS[x])$ as the generic form for a sum over histories in configuration space is discussed critically and placed in its proper context. The standard derivation of the sum over paths by discretizing the paths is reviewed, and it is shown that the form $\sum \exp(iS[x])$ is justified only for Schrodinger-type systems which are at most second order in the momenta. Extending this derivation to the relativistic free particle, the causal Green's function is expressed as a sum over timelike paths, and the Feynman Green's function is expressed both as a sum over paths which only go one way in time and as a sum over paths which move forward and backward in time. The weighting of the paths is shown not to be $\exp(iS[x])$ in any of these cases. The role of the inner product and the operator ordering of the wave equation in defining the sum over histories is discussed.