  This paper extends the formalism for quantizing field theories via a
microcanonical quantum field theory and Hamilton's principle to classical
evolution equations. These are based on the well-known correspondence under a
Wick rotation between quantum field theories and 4-D statistical mechanical
theories. By placing quantum field theories on a 4+1-D under Wick rotation to
5-D, expectations of observables are calculated for a microcanonical field
theory averaging Hamiltonian flow over a fifth spacelike dimension, a technique
common in lattice gauge simulations but not in perturbation theory. In a novel
demonstration, averaging pairs of external lines in the classical Feynman
diagrams over the fifth dimension generates diagrams with loops and vacuum
fluctuations identical to Standard Model diagrams. Because it is
microcanonical, this approach, while equivalent for standard quantum fields
theories in the Standard Model, is able to quantize theories that have no
canonical quantization. It is also unique in representing expectations as
averages over solutions to an ordinary, classical PDE rather than a path
integral or operator based approaches. Hence, this approach draws a clear
connection between quantum field theory and classical field theory in higher
dimensions which has implications towards how quantum effects are interpreted.
In particular, it raises questions about how violations of the ergodic
hypothesis could influence quantum measurements even in standard,
non-statistical quantum field theory.
