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

Abstract

A set of Green functions ${\mathit{scrG}}_{\mathrm{\ensuremath{\alpha}}}$(x-y), \ensuremath{\alpha}\ensuremath{\in}[0,2\ensuremath{\pi} [for free scalar field theory is introduced, varying between the Hadamard Green function ${\mathrm{\ensuremath{\Delta}}}_{1}$(x-y)==〈0\ensuremath{\Vert}{cphi(x),cphi(y)}\ensuremath{\Vert}0〉 and the causal Green function ${\mathit{scrG}}_{\mathrm{\ensuremath{\pi}}}$(x-y)=i\ensuremath{\Delta}(x-y)==[cphi(x),cphi(y)]. For every \ensuremath{\alpha}\ensuremath{\in}[0,2\ensuremath{\pi}[ a path integral representation for ${\mathit{scrG}}_{\mathrm{\ensuremath{\alpha}}}$ is obtained both in configuration space and in the phase space of the classical relativistic particle. Setting \ensuremath{\alpha}=\ensuremath{\pi} a sum-over-histories representation for the causal Green function is obtained. Furthermore, a reduced phase space integral representation for the ${\mathit{scrG}}_{\mathrm{\ensuremath{\alpha}}}$'s is stated and an alternative BRST path integral representation for ${\mathit{scrG}}_{\mathrm{\ensuremath{\alpha}}}$ is presented. From these path integral representations the composition laws for the ${\mathit{scrG}}_{\mathrm{\ensuremath{\alpha}}}$'s are derived using a modified path decomposition expansion.