Quantization and superselection sectors III: Multiply connected spaces and indistinguishable particles

TL;DR

This paper uses C*-algebraic deformation quantization to unify different approaches to quantum indistinguishable particles, showing that parastatistics can be realized through internal degrees of freedom within bosons and fermions.

Contribution

It demonstrates that topologically nontrivial configuration spaces correspond to observable algebras quantized via operator methods, reconciling different formalisms and clarifying the nature of parastatistics.

Findings

Parastatistics can be realized with internal degrees of freedom.

Configuration space topology relates to observable algebra structure.

Particle states with parastatistics are equivalent to bosons/fermions with internal symmetries.

Abstract

We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel's notion of C*-algebraic (`strict') deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach due to Souriau (1967), Laidlaw and DeWitt (1971), Leinaas and Myrheim (1977), and others. The former allows parastatistics, whereas the latter apparently leaves room for bosons and fermions only. This seems to contradict the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that the topologically nontrivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Second, we show that the irreducible representations of the latter may be realized by vector bundle constructions, which include the line bundles of the second approach: representations on higher-dimensional bundles (which define parastatistics) cannot be excluded a priori. However, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom. Although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher-Haag-Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to QCD.