  We show that non-relativistic Quantum Mechanics can be faithfully represented
in terms of a classical diffusion process endowed with a gauge symmetry of
group Z_4. The representation is based on a quantization condition for the
realized action along paths. A lattice regularization is introduced to make
rigorous sense of the construction and then removed. Quantum mechanics is
recovered in the continuum limit and the full U(1) gauge group symmetry of
electro-magnetism appears. Anti-particle representations emerge naturally,
albeit the context is non-relativistic. Quantum density matrices are obtained
by averaging classical probability distributions over phase-action variables.
We find that quantum conditioning can be described in classical terms but not
through the standard notion of sub sigma-algebras. Delicate restrictions arise
by the constraint that we are only interested in the algebra of gauge invariant
random variables. We conclude that Quantum Mechanics is equivalent to a theory
of gauge invariant classical stochastic processes we call Stochastic Mechanics.
