  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.
