Spin-Statistics Theorem and Geometric Quantisation

Abstract

We study how the spin-statistics theorem relates to the geometric structures on phase space that are introduced in quantisation procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the non-relativistic domain (in fact for any symmetry group including internal symmetries) without quantum field theory, by the requirement that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates. We then examine our construction in the geometric and the coherent-state-path-integral quantisation schemes. In the appendix we apply our results to exotic systems exhibiting continuous ``spin'' and ``fractional statistics''. This gives novel and unusual forms of the spin-statistics relation.