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

Abstract

A set of Green functions ${\mathcal{G}}_{\alpha }(x-y),\alpha \in [0,2\pi [$, 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 ${\mathcal{G}}_{\pi }(x-y)=i\Delta (x-y)\equiv [\phi (x),\phi (y)]$. For every $\alpha \in [0,2\pi [$ a path-integral representation for ${\mathcal{G}}_{\alpha }$ 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 ${\mathcal{G}}_{\alpha }$ is presented. From these path integral representations the composition laws for the ${\mathcal{G}}_{\alpha }$'s are derived using a modified path decomposition expansion.

14 pages, latex, DESY 93-145

References in corpus (3)

• Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology
• Gauge Invariance for Generally Covariant Systems
• The use of exp(iS[x]) in the sum over histories

Cited by in corpus (2)

• Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts
• Quantization of simple parametrized systems