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Rieffel induction as generalized quantum Marsden-Weinstein reduction

N. P. Landsman

TL;DR

This work reframes generalized symplectic reduction as a classical induction process and identifies Rieffel induction as its natural quantum analogue, linking symplectic imprimitivity to Morita equivalence of Poisson manifolds. By replacing symplectic groupoids with operator-algebraic techniques, the authors derive a broad quantum reduction framework applicable to compact, amenable, and groupoid settings, including non-strongly Hamiltonian actions via cocycles and twisted algebras. The paper develops the machinery of rigging maps and induced representations, illustrating with coadjoint-orbit, semidirect-product, Yang–Mills, relativistic, and finite $W$-algebra examples how constrained systems quantize cleanly without BRST obstructions. It unifies classical and quantum reduction under Morita equivalence, providing explicit constructions for a variety of constrained systems and indicating pathways to infinite-dimensional generalizations. Overall, the approach offers a conceptually transparent, computationally tractable route from generalized reduction to quantum representations.

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 operator-valued inner product on a (pre-) Hilbert space (that is, a structure similar to a Hilbert $C^*$-module). Hence Rieffel's far-reaching operator-algebraic 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.

Rieffel induction as generalized quantum Marsden-Weinstein reduction

TL;DR

This work reframes generalized symplectic reduction as a classical induction process and identifies Rieffel induction as its natural quantum analogue, linking symplectic imprimitivity to Morita equivalence of Poisson manifolds. By replacing symplectic groupoids with operator-algebraic techniques, the authors derive a broad quantum reduction framework applicable to compact, amenable, and groupoid settings, including non-strongly Hamiltonian actions via cocycles and twisted algebras. The paper develops the machinery of rigging maps and induced representations, illustrating with coadjoint-orbit, semidirect-product, Yang–Mills, relativistic, and finite -algebra examples how constrained systems quantize cleanly without BRST obstructions. It unifies classical and quantum reduction under Morita equivalence, providing explicit constructions for a variety of constrained systems and indicating pathways to infinite-dimensional generalizations. Overall, the approach offers a conceptually transparent, computationally tractable route from generalized reduction to quantum representations.

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 operator-valued inner product on a (pre-) Hilbert space (that is, a structure similar to a Hilbert -module). Hence Rieffel's far-reaching operator-algebraic 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 -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.

Paper Structure

This paper contains 17 sections, 11 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Symplectic induction
  3. Generalized Marsden-Weinstein reduction
  4. Symplectic imprimitivity theorem
  5. Symplectic induction in stages
  6. Quantization of the symplectic induction procedure
  7. Rieffel induction
  8. Quantum Marsden-Weinstein reduction
  9. Quantization of symplectic group actions which are not strongly Hamiltonian
  10. Induction with groupoid algebras
  11. Some examples
  12. Co-adjoint orbits and unitary representations of semidirect products
  13. Co-adjoint orbits and unitary representations of nilpotent Lie groups
  14. The generalized Yang-Mills construction
  15. The illusion of time
  16. ...and 2 more sections

Key Result

Theorem 1

$S*_{P} S_{\rho}$ is co-isotropically immersed in $S\times S_{\rho}$. The collection of vector fields $\{ \hat{X}_f| f\in C^{\infty}(P)\}$ defines a (generally singular) foliation ${\cal F}$, of $S*_{P} S_{\rho}$, whose leaf space $S^{\rho}=S*_{P} S_{\rho}/{\cal F}$ coincides with the quotient of $S

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • ...and 3 more