The BRS method and geometric quantization: Some examples

Duval, Christian; Elhadad, J.; Tuynman, Gijs M.

doi:10.1007/bf02125699

Cited by 26 publications

(18 citation statements)

References 19 publications

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“…Let L 2 (M, E) N be the Hilbert space constructed from L 2 (M, E) as in the definition of the homomorphism V N . We will show that L 2 (M, E) N is naturally isomorphic 19 to the Hilbert space…”

Section: Spaces Of L 2 -Sectionsmentioning

confidence: 99%

The Guillemin–Sternberg conjecture for noncompact groups and spaces

Hochs¹,

Landsman²

2008

J. K-Theory

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The Guillemin-Sternberg conjecture states that "quantisation commutes with reduction" in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spin c Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin-Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the "quantisation commuates with reduction" phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C * -algebra C * (G) in a crucial way. For example, the equivariant index -which in the compact case takes values in the representation ring R(G) -is replaced by the analytic assembly map -which takes values in K0(C * (G)) -familiar from the Baum-Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.1 Here the motivating role played by two closely related papers on the discrete series representations of semisimple Lie groups should be mentioned. It seems that Parthasarathy [55] influenced Bott's formulation of the Guillemin-Sternberg conjecture (see below), whereas Atiyah and Schmid [3] in part inspired the Baum-Connes conjecture (see [12,36]).

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Section: Spaces Of L 2 -Sectionsmentioning

confidence: 99%

The Guillemin–Sternberg conjecture for noncompact groups and spaces

Hochs¹,

Landsman²

2008

J. K-Theory

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“…For example, the infinitesimal Blattner-Kostant-Sternberg pairing of real polarizations leads to c = − 1 6 [14,16]. In the case of n-dimensional spheres, the pairing of complex polarizations gives c = − n−1 4n [12] while Weyl quantization and quantum reduction would lead to c = n+1 4n (see [3] for a survey).…”

Section: A Few Ideas Around Quantizationmentioning

confidence: 99%

Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

Duval

Ovsienko

1997

Advances in Mathematics

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“…[15,13]). For the moment, we just wish to point out that serious difficulties of principle with the latter were spelled out in [7,19], and that on the practical side "at present the computation of BRST-cohomology is an extremely difficult problem" [13]. Moreover, the Rieffel induction process mimics the symplectic procedure more closely than any BRST treatment we are aware of (including the bosonic BRST theory in [6]), and appears to be simpler both conceptually and computationally.…”

Section: It Was Recognized Bymentioning

confidence: 99%

Rieffel induction as generalized quantum Marsden-Weinstein reduction

Landsman

1995

Journal of Geometry and Physics

101

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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 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. * Supported by an S.E.R.C. Advanced Research Fellowship

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The BRS method and geometric quantization: Some examples

Abstract: We recall the method of quantization of constraints as outlined by Kostant and Sternberg. We discuss various examples to test compatibility between geometric quantization and this procedure. In almost all cases reduction and quantization commute.

Cited by 26 publications

References 19 publications

The Guillemin–Sternberg conjecture for noncompact groups and spaces

The Guillemin–Sternberg conjecture for noncompact groups and spaces

Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

Rieffel induction as generalized quantum Marsden-Weinstein reduction

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