Phase space reduction for star-products: An explicit construction for \(\mathbb{C} P^ n\). (English) Zbl 0849.58035

Bordemann, M.; Brischle, M.; Emmrich, C.; Waldmann, S.

Lett. Math. Phys. 36, No. 4, 357-371 (1996).

The authors give an explicit formula for a star-product on the complex projective space and the domain \(SU(1,n)/SU(1) \times U(n)\) which in addition is shown to be the Marsden-Weinstein reduction of a star product equivalent to the usual Wick product on the flat Kähler manifold \(\mathbb{C}^{n + 1} \backslash \{0\}\).

Reviewer: M.Puta (Timişoara)

MSC:

53D50	Geometric quantization
53B35	Local differential geometry of Hermitian and Kählerian structures
81S10	Geometry and quantization, symplectic methods

Keywords:

star-product; Marsden-Weinstein reduction; Wick product

References:

[1]	AbrahamR. and MarsdenJ. E.,Foundations of Mechanics, 2nd edn, Addison-Wesley, Reading, Mass., 1985.
[2]	BayenF., FlatoM., FronsdalC., LichnerowiczA. and SternheimerD., Deformation theory and quantization.Ann. of Physics 111 (1978), part I: 61-110, part II: 111-151. · Zbl 0377.53024
[3]	BerezinF., Quantization,Izv. Mat. Nauk 38 (1974), 1109-1165.
[4]	Bordemann, M., Meinrenken, E. and Römer, H., Total space quantization of Kähler manifolds, preprint Univ. Freiburg THEP 93/5 (revised version to appear).
[5]	CahenM., DeWildeM. and GuttS., Local cohomology of the algebra ofC ?-functions on a connected manifold,Lett. Math. Phys. 4 (1980), 157-167. · Zbl 0453.58026
[6]	CahenM., GuttS. and RawnsleyJ., Quantization of Kähler manifolds I,J. Geom. Phys. 7 (1990), 45-62. · Zbl 0736.53056
[7]	CahenN., GuttS. and RawnsleyJ., Quantization of Kähler manifolds II,Trans. Amer. Math. Soc. 337 (1993), 73-98. · Zbl 0788.53062
[8]	CahenM., GuttS. and RawnsleyJ., Quantization of Kähler manifolds III,Lett. Math. Phys. 30 (1994), 291-305. · Zbl 0826.53052
[9]	DeWildeM. and LecomteP. B. A., Existence of star-products and of formal deformations of the Poisson Lie Algebra of arbitrary symplectic manifolds,Lett. Math. Phys. 7 (1983), 487-496. · Zbl 0526.58023
[10]	Fedosov, B., Formal quantization, inSome Topics of Modern Mathematics and Their Applications to Problems of Mathematical Physics, Moscow, 1985, pp. 129-136.
[11]	FedosovB., A simple geometrical construction of deformation quantization,J. Differential Geom. 40 (1994), 213-238. · Zbl 0812.53034
[12]	FedosovB., Reduction and eigenstates in deformation quantization, in Demuth, Schrohe, Schulze (eds),Pseudodifferential Calculus and Mathematical Physics, Academiæ Verlag, Berlin, 1994; Preprint: Moscow Institute of Science and Technology, 1993.
[13]	GerstenhaberM. and SchackS. D., Algebraic cohomology and deformation theory, in M.Hazewinkel and M.Gerstenhaber (eds.),Deformation Theory of Algebras and Structures and Applications, Kluwer Acad. Publ. Dordrecht, 1988.
[14]	GriffithsP. and HarrisJ.,Principles of Algebraic Geometry, Wiley, New York, 1978.
[15]	GuttS., An explicit *-product on the cotangent bundle of a Lie group,Lett. Math. Phys. 7 (1983), 249-258. · Zbl 0522.58019
[16]	HelgasonS.,Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
[17]	MorenoC. and Ortega-NavarroP., *-products on 371-1,S 2 and related spectral analysis,Lett. Math. Phys. 7 (1983), 181-193. · Zbl 0528.58014
[18]	MorenoC., *-products on some Kähler manifolds,Lett. Math. Phys. 11 (1986), 361-372. · Zbl 0618.53049
[19]	MorenoC., Geodesic symmetries and invariant star products on Kähler symmetric spaces,Lett. Math. Phys. 13 (1987), 245-257. · Zbl 0628.58018