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

Abstract

A set of Green functions ${\cal G}_{\alpha}(x-y), \alpha \in [0, 2 \pi [$, for free scalar field theory is introduced, varying between the Hadamard Green function $\Delta_1(x-y) \equiv \linebreak[2] \lsta{0} \hspace{-0.1cm} \{ \varphi(x), \varphi(y) \} \hspace{-0.1cm} \rsta{0}$ and the causal Green function ${\cal G}_{\pi}(x-y) = i \Delta(x-y) \equiv [\varphi(x), \varphi(y)]$. For every $\alpha \in [0, 2 \pi [$ a path-integral representation for ${\cal G}_{\alpha}$ is obtained both in the configuration space and in the phase space of the classical relativistic particle. Especially setting $\alpha = \pi$ a sum-over-histories representation for the causal Green function is obtained. Furthermore using BRST theory an alternative path-integral representation for ${\cal G}_{\alpha}$ is presented. From these path integral representations the composition laws for the ${\cal G}_{\alpha}$'s are derived using a modified path decomposition expansion.