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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

##### MSC:
 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
##### Keywords:
geometric quantization
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##### References:
  [Bla1] Blattner, R. J.: Pairing of half-form spaces. In: Coll. Int. CNRS no237. Souriau, J. M. (ed.). pp. 175–186 Aix en Provence: CNRS 1974  [Bla2] –: The metalinear geometry of non-real polarizations, Lecture Notes in Math., Vol.570, pp. 11–45. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0346.53027  [BRS] Becchi, C., Rouet, A., Stora, R.: The abelian Higgs Kibble model, unitarity of theS-operator, Phys. Lett.B52, 344–346 (1974); Renormalization of gauge theories, Ann. Phys. (NY)98, 287–321 (1976)  [Dir] Dirac, P. A. M.: Lectures on quantum mechanics, Belfer Graduate School of Science, Yeshiva University, New York (1964)  [Dub] Dubois Violette, M.: Systèmes dynamiques contraints: l’approche homologique. Ann. Inst. Fourier37, 45–57 (1987) · Zbl 0635.58007  [DET] Duval, C., Elhadad, J., Tuynman, G. M.: Hyperfine interaction in a classical hydrogen atom and geometric quantization. J. Geom. Phys.3, 401–420 (1986) · Zbl 0615.58024 · doi:10.1016/0393-0440(86)90015-X  [Elh] Elhadad, J.: Quantification du flot géodésique de la sphère Sn, C.R.. Acad. Sci. Paris285 961–964 (1977) · Zbl 0403.58002  [FHST] Fisch, J., Henneaux, M., Stasheff, J., Teitelboim, C.: Existence, uniqueness and cohomology of the classical BRST charge with ghosts of ghosts, Commun. Math. Phys.120, 379–407 (1988) · Zbl 0685.58054 · doi:10.1007/BF01225504  [FV] Fradkin, E. S., Vilkoviski, G. A.: Quantization of relativistic systems with constraints, equivalence of canonical and covariant formalisms in quantum theory of gravitational field CERN Report Th. 2332 (1977)  [Got] Gotay, M. J.: Constraints, reduction, and quantization. J. Math. Phys.27, 2051–2066 (1986) · Zbl 0632.58020 · doi:10.1063/1.527026  [GS] Guillemin, V., Sternberg, S.: Geometric Asymptotics. Providence, RI: American Mathematical Society 1977 · Zbl 0364.53011  [HT] Henneaux, M., Teitelboim, C.: BRST cohomology in classical mechanics. Commun. Math. Phys.115, 213–230 (1988) · Zbl 0649.58050 · doi:10.1007/BF01466770  [Kos] Kostant, B.: Quantization and unitary representations. Lecture Notes in Mathematics, vol.170, pp. 87–208 Berlin, Heidelberg, New York: Springer 1970  [KS] Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys. (NY)176, 49–113 (1987) · Zbl 0642.17003 · doi:10.1016/0003-4916(87)90178-3  [Lol] Loll, R.: The extended phase space of the BRS approach. Commun. Math. Phys.119, 509–527 (1988) · Zbl 0684.58049 · doi:10.1007/BF01218085  [MW] Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121–130 (1974) · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4  [Mos1] Moser, J.: Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. Math.23, 609–636 (1970) · Zbl 0193.53803 · doi:10.1002/cpa.3160230406  [Mos2] –: Various aspects of integrable Hamiltonian systems. In: Dynamical Systems. Guckenheimer, J., Moser, J., Newhouse, S. E. (eds.) Progress In Math. Vol.8, Boston, MA: Birkhauser 1980 · Zbl 0468.58011  [NR] Nielsen, H. B., Rohrlich, D.: A path integral to quantize spin, Nucl. Phys.B299, 471–483 (1988) · doi:10.1016/0550-3213(88)90545-7  [Raw] Rawnsley, J.: A nonunitary pairing of polarizations for the Kepler Problem. Trans. A.M.S.250, 167–178 (1979) · Zbl 0422.58019 · doi:10.1090/S0002-9947-1979-0530048-1  [Sni] Sniatycki, J.: Geometric quantization and quantum mechanics. Appl. Math. Sci. Vol.50, Berlin, Heidelberg, New York: Springer 1980 · Zbl 0429.58007  [Sou1] Souriau, J. M.: Structure des systèmes dynamiques. Paris: Dunod 1969  [Sou2] Souriau, J. M.: Sur la variété Kepler. In: Symposia Mathematica XIV (1974), pp. 343–360; Géométrie globale du problème à deux corps. In: Proceedings IUTAM-ISIMM Modern Developments in Analytical Mechanics, Atti della Accademia delle Scienze di Tornio. Supp. al Vol.117, 369–418 (1983)  [Sta] Stasheff, J.: Constrained poisson algebras and strong homotopy representation. Bull. AMS19, 287–290 (1988) · Zbl 0669.18009 · doi:10.1090/S0273-0979-1988-15645-5  [Tuy1] Tuynman, G. M.: Generalized Bergman kernels and geometric quantization. J. Math. Phys.28, 573–583 (1987) · Zbl 0616.58041 · doi:10.1063/1.527642  [Tuy2] -Tuynman, G. M.: Reduction, quantization and non-unimodular groups. J. Math. Phys.31, (1990)  [Woo] Woodhouse, N.: Geometric quantization. Oxford: Oxford University Press 1980 · Zbl 0458.58003
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