  Quantum Yang-Mills theory, Classical Statistical Field Theory (for
Hamiltonians which are non-polynomial in the fields, e.g. General relativistic
statistical mechanics) and Quantum Gravity all suffer from severe mathematical
inconsistencies and produce unreliable predictions at best.
  We define with mathematical rigor, a class of statistical field theories in
Minkowski space-time where the (classical) canonical coordinates when modified
by a non-deterministic time evolution, verify the canonical commutation
relations. We then extend these statistical field theories to include
non-trivial gauge symmetries and show that these theories have all the features
of a Quantum Yang-Mills theory in four-dimensional space-time. We generalize
the Gaussian measure to allow for the definition of Hamiltonians which are
non-polynomial in the fields, such as in Classical Statistical Field Theory and
Quantum Gravity.
  Finally, we test the consistency of our formalism with the quantization of
the free Electromagnetic field.
