The geometric phase and the spin-statistics relation

Abstract

The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson's stochastic mechanics. This introduces a direction vector attached to a space-time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry-Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the 'exchange rotation' in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase. © 2010 The Royal Society.