 
Summary: STOCHASTIC MECHANICS AS A GAUGE THEORY
CLAUDIO ALBANESE
Abstract. We show that nonrelativistic Quantum Mechanics can be faithfully represented
in terms of a classical diffusion process endowed with a gauge symmetry of group Z4. The
representation is based on a quantization condition for the realized action along paths. A lat
tice 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 sym
metry of electromagnetism appears. Antiparticle representations emerge naturally, albeit
the context is nonrelativistic. Quantum density matrices are obtained by averaging classical
probability distributions over phaseaction variables. We find that quantum conditioning can
be described in classical terms but not through the standard notion of sub algebras. Del
icate 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.
Contents
1. Quantization Condition 2
2. Lattice Regularization 4
3. The Joint Process for the Position and the Realized Action 5
4. Density Matrices 6
5. Classical and Quantum Conditioning 8
