The BRS method of geometric quantization: Some examples
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 \(D=d+2(-1)^ p(1\otimes \delta).\) Then if \(H_ q(K)=0\) for any \(q>0\) it results \(C^{\infty}(B)\simeq H^ 0({\mathcal G},H_ 0(K))\simeq H^ 0_ D(L)\). Following the KS-procedure \(L=\Lambda {\mathcal G}^*\otimes \Lambda {\mathcal G}\otimes C^{\infty}(M)\) an be endowed with a structure of super Poisson algebra. Further identifications are given, involving some elements \(\theta\) and Td of L. 2. BRS Quantization. One considers the quantizations of the algebras \(C^{\infty}(B)\) and L, by constructing a linear map from a subspace of \(C^{\infty}(M)\) to the (skew)self-adjoint operators on a Hilbert space \({\mathcal H}_ M\). The KS approach to the quantization of L consists in quantizing L as operators on \({\mathcal H}_ L=(\Lambda {\mathcal G}^*)^ C\otimes {\mathcal H}_ M\) by the ordinary quantization \(\tau_ 0: C^{\infty}(M)\to End{\mathcal H}_ M\) of \(C^{\infty}(M)\). The next paragraphs give some explicit examples. 3. The geodesic flow and boosts on spheres. Now the reduced symplectic manifold is \(B=T^*S^ n\). One quantizes different observables related to a geodesic flow as for instance the Hamiltonian \(h_ 1=\| p\|^ 2\) (the kinetic energy). 4. The Kepler manifold or the null geodesics on compactified Minkowski space-time. Now \(B=\{\mu \in o^*(4,2)\) \(\mu^ 2=0\}\) is a co-adjoint orbit of O(4,2) and \(B=B_+\cup B_-\) where \(B_{\pm}\simeq \dot T(S^ 3)\). The symplectic reduction is via SL(2,R). 5. Reduction by a non-unimodular group. The cotangent bundle \(B=T^*S^ n\) can be obtained by a classical reduction from \(M=T(R^{n+1,1}\{0\})\). Concerning the quantum reduction by vertical polarization one can quantize directly all observables which are linear in J. 6. The spin. The spin phase space \((B,\omega_ n)=(S^ 2,s\cdot Surf)\) is handled by a reduction with group U(1) where \(s>0\) is a constant (the scalar spin) and Surf is the Riemannian surface element of \(S^ 2\subset R^ 3\). The quantum reduction is considered. One gives the BRS-KS- quantization of \(s_ 3=<s,e_ 3>\) of \(S^ 2\). The spin squared and the spin-spin intersection are considered. 7. Conclusions and final remarks. The compatibility between geometric quantization and the KS-procedure is generally favourable. However there remain several unsolved problems, concerning the sufficient conditions for this, especially the case when \(0\in {\mathcal G}^*\) is a regular value for \(\psi\).
scientific article; zbMATH DE number 4147154
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