The BRS method of geometric quantization: Some examples
type: Item
instance of: scholarly article
title: The BRS method of geometric quantization: Some examples
zbMATH Open document ID: 0699.58040
DOI: 10.1007/BF02125699
author: J. Elhadad
author: Christian Duval
author: Gijs Tuynman
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: The paper provides a synthesis of some general schemes for quantization of a reduced system going from the BRS procedure [see \textit{C. Becchi}, \textit{A. Rouet} and \textit{R. Stora}, Ann. Phys. 98, 287-321 (1976)] and from the KS-method of quantization of the constraints [see \textit{B. Kostant} and \textit{S. Sternberg}, Ann. Phys. 176, 49-113 (1987; Zbl 0642.17003)]. In the introduction the authors present their motivation to test the compatibility between the geometric quantization and this method. The paragraphs of the paper are the following: 1. The classical BRS symmetry. Let (M,\(\omega)\) be a symplectic manifold, G a connected Lie group, \({\mathcal G}\) its Lie algebra, \(\phi\) : \(G\to Diff M\) a symplectic action of G admitting a momentum mapping \(\psi\) : \(M\to {\mathcal G}^*\) with J: \({\mathcal G}\to C^{\infty}(M)\). Let \(0\in {\mathcal G}^*\) be a weakly regular value of \(\psi\) so that \(C=\psi^{-1}(0)\) is a closed submanifold of M. Define \(B=C/G\) and assume that \(\pi\) : \(C\to B\) is smooth. Consider \(C^{\infty}(M)\), \(C^{\infty}(B)\) as Poisson algebras and let I be the ideal \(I={\mathcal I}({\mathcal G})C^{\infty}(M)\). There are the identifications \(C^{\infty}(C)\simeq C^{\infty}(M)/I\), \(C^{\infty}(B)\simeq \{f\in C^{\infty}(M)|\) \(\forall \xi \in {\mathcal G}\), \(\{\) J(\(\xi)\),f\(\}\in I\}/I\). The result above can be translated in co(homological) terms considering \(K=\Lambda {\mathcal G}\otimes C^{\infty}(M)\) as a Koszul-complex with the boundary operator \(\delta\) defined in terms of J. One considers the cohomology H(\({\mathcal G},E)\) of \({\mathcal G}\) with values in the \({\mathcal G}\)-module E, as well as the complex (L,d), \(L=\Lambda {\mathcal G}^*\otimes E\). By a suitable representation \(\rho\) of \({\mathcal G}\) on K, K becomes a \({\mathcal G}\)-module. Using the theory of spectral sequences one considers \(L=\Lambda {\mathcal G}^*\otimes K\) as a complex (L,D) with the operator \(D=d+2(-1)^ p(1\otim…
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 4147154
zbMATH Keywords: geometric quantization
MaRDI profile type: Publication
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Wikidata QID: Q125622161
description: scientific article; zbMATH DE number 4147154