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

A set of Green functions Gα(x − y), α ∈ [0, 2π[, for free scalar field theory is introduced, varying between the Hadamard Green function ∆1(x − y) ≡ 〈0 | {φ(x), φ(y)} | 0〉 and the causal Green function Gπ(x − y) = i∆(x − y) ≡ [φ(x), φ(y)]. For every α ∈ [0, 2π[ a path-integral representation for Gα 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 Gα is presented. From these path integral representations the composition laws for the Gα’s are derived using a modified path decomposition expansion. ∗ Supported by Deutsche Forschungsgemeinschaft