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The BRS method of geometric quantization: Some examples. (English) Zbl 0699.58040
The paper provides a synthesis of some general schemes for quantization of a reduced system going from the BRS procedure [see C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98, 287-321 (1976)] and from the KS-method of quantization of the constraints [see B. Kostant and 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 \(\psi\).
Reviewer: M.Tarina

53D50 Geometric quantization
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
81S10 Geometry and quantization, symplectic methods
Full Text: DOI
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