  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. This is advantageous, in particular for gauge theories,
because it allows a different type of gauge fixing that avoids the Gribov
problem. Traditionally, in this approach, the fictitious 5th time is the analog
of computer time in a Monte Carlo simulation of 4-dimensional Euclidean fields.
A Euclidean probability distribution which depends on the 5th time relaxes to
an equilibrium distribution. 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.
