The quantization of constrained systems: from symplectic reduction to Rieffel induction

TL;DR

This paper introduces a generalized symplectic reduction method for constrained systems and demonstrates its quantization via Rieffel induction, connecting symplectic geometry with operator algebra theory and illustrating with detailed examples.

Contribution

It generalizes the Marsden-Weinstein reduction and shows how to quantize it using Rieffel induction with C*-modules, linking geometric and algebraic approaches.

Findings

Successfully quantized generalized symplectic reductions

Connected constrained quantization with Rieffel induction and C*-modules

Provided detailed examples illustrating the quantization process

Abstract

This is an introduction to the author's recent work on constrained systems. Firstly, a generalization of the Marsden-Weinstein reduction procedure in symplectic geometry is presented - this is a reformulation of ideas of Mikami-Weinstein and Xu. Secondly, it is shown how this procedure is quantized by Rieffel induction, a technique in operator algebra theory. The essential point is that a symplectic space with generalized moment map is quantized by a pre-(Hilbert) C^*-module. The connection with Dirac's constrained quantization method is explained. Three examples with a single constraint are discussed in some detail: the reduced space is either singular, or defined by a constraint with incomplete flow, or unproblematic but still interesting. In all cases, our quantization procedure may be carried out. Finally, we re-interpret and generalize Mackey's quantization on homogeneous spaces. This provides a double illustration of the connection between C^*-modules and the moment map.