From stochastic quantization to bulk quantization: Schwinger-Dyson equations and S-matrix

In stochastic quantization, ordinary 4-dimensional Euclidean quantum field theory is expressed as a functional integral over fields in 5 dimensions with a fictitious 5th time. However a broader framework, which we call "bulk quantization", is required for extension to fermions, and for the increased power afforded by the higher symmetry of the 5-dimensional action that is topological when expressed in terms of auxiliary fields. Within the broader framework, we give a direct proof by means of Schwinger-Dyson equations that a time-slice of the 5-dimensional theory is equivalent to the usual 4-dimensional theory. The proof does not rely on the conjecture that the relevant stochastic process relaxes to an equilibrium distribution. Rather, it depends on the higher symmetry of the 5-dimensional action which includes a BRST-type topological invariance, and invariance under translation and inversion in the 5-th time. We express the physical S-matrix directly in terms of the truncated 5-dimensional correlation functions, for which "going off the mass-shell" means going from the 3 physical degrees of freedom to 5 independent variables. We derive the Landau-Cutokosky rules of the 5-dimensional theory which include the physical unitarity relation.