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STOCHASTIC MECHANICS AS A GAUGE THEORY CLAUDIO ALBANESE
 

Summary: STOCHASTIC MECHANICS AS A GAUGE THEORY
CLAUDIO ALBANESE
Abstract. 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 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 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 -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

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics