Sum-over-histories representation for the causal Green function of free scalar field theory

Abstract

A set of Green functions ${\cal G}_α(x-y), α\in [0, 2 π[$, for free scalar field theory is introduced, varying between the Hadamard Green function $Δ_1(x-y) \equiv \linebreak[2] \lsta{0} \hspace{-0.1cm} \{ φ(x), φ(y) \} \hspace{-0.1cm} \rsta{0}$ and the causal Green function ${\cal G}_π(x-y) = i Δ(x-y) \equiv [φ(x), φ(y)]$. For every $α\in [0, 2 π[$ a path-integral representation for ${\cal G}_α$ is obtained both in the configuration space and in the phase space of the classical relativistic particle. Especially setting $α= π$ a sum-over-histories representation for the causal Green function is obtained. Furthermore using BRST theory an alternative path-integral representation for ${\cal G}_α$ is presented. From these path integral representations the composition laws for the ${\cal G}_α$'s are derived using a modified path decomposition expansion.