 93289 N.P. Landsman
 Rieffel induction as generalized quantum MarsdenWeinstein reduction
(119K, LaTeX)
Nov 10, 93

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. A new approach to the quantization of constrained or otherwise reduced classical mechanical systems
is proposed. On the classical side, the generalized symplectic reduction procedure of Mikami and
Weinstein, as further extended by Xu in connection with symplectic equivalence bimodules and Morita
equivalence of Poisson manifolds, is rewritten so as to avoid the use of symplectic groupoids, whose
quantum analogue is unknown. A theorem on symplectic reduction in stages is given. This allows one to
discern that the `quantization' of the generalized moment map consists of an operatorvalued inner
product on a (pre) Hilbert space (that is, a structure similar to a Hilbert $C^*$module). Hence
Rieffel's farreaching operatoralgebraic generalization of the notion of an induced representation
is seen to be the exact quantum counterpart of the classical idea of symplectic reduction, with
imprimitivity bimodules and strong Morita equivalence of $C^*$algebras falling in the right place.
Various examples involving groups as well as groupoids are given, and known difficulties with both
Dirac and BRST quantization are seen to be absent in our approach.
 Files:
93289.tex