Landsman, N.P. (Klaas)
(2013)
Quantization and superselection sectors III: Multiply connected spaces and indistinguishable particles.
[Preprint]
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
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 to
the heuristic arguments which led Gell-Mann and others to QCD.
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