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STOCHASTIC MECHANICS AS A GAUGE THEORY CLAUDIO ALBANESE
 

Summary: STOCHASTIC MECHANICS AS A GAUGE THEORY
CLAUDIO ALBANESE
Abstract. We introduce a classical diffusion process which provides a full description of
non-relativistic Quantum Mechanics and has the form of a Z4 gauge theory. We first define
a stochastic process on a discretization of physical space of the form (aZ)3, where a is an
elementary length scale. We then lift this process to the principal bundle (aZ)3 Z4. Non-
relativistic Quantum Mechanics is recovered in the limit as a 0, as we show in the case of
a scalar particle in an electromagnetic field. Many-body interactions can easily be accommo-
dated. In the case of tight binding Hamiltonians no limit needs to be taken, the equivalence is
straightforward and sheds new light on the dynamics of quantum phases in solid state Physics.
The physical interpretation of Quantum Mechanics in the continuum limit reveals subtle dif-
ferences between Quantum and Classical Probability and provides an intriguing geometric
explanation of quantum coherence and a link to gauge theories.
A question that attracted much attention is whether the Schrodinger equation can be in-
terpreted as a classical diffusion. The first attempt in this direction is in (Fenyes 1952). This
was then greatly expanded upon in (Nelson 1967) and became known as Nelson's Stochastic
Mechanics. See also (de la Pena-Auerbach 1970), (Jammer 1974), (Guerra and Ruggiero 1973).
These efforts were hampered by the difficulty of accommodating many-body interactions while
avoiding Bell's inequalities, see (Bell 1966), (Aspect et al. 1982). In fact, there appear to be
irreconcilable differences between Classical and Quantum Probability. Quantum entanglement

  

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

 

Collections: Mathematics