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**Homogeneous Fedosov star products on cotangent bundles. II: GNS representations, the WKB expansion, traces, and applications.**
*(English)*
Zbl 0989.53060

Summary: This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold \(Q\). For certain homogeneous star products of Weyl ordered type which we have obtained from a Fedosov type procedure in part I, see [Commun. Math. Phys. 198, 363-396 (1998; Zbl 0968.53056)] we construct differential operator representations via the formal GNS construction [see Martin Bordemann and Stefan Waldmann [Commun. Math. Phys. 195, No.3, 549-583 (1998; Zbl 0989.53057)]. The positive linear functional is integration over \(Q\) with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furthermore we discuss simple examples like free particle Hamiltonians (defined by a Riemannian metric on \(Q)\) and the implementation of certain diffeomorphisms of \(Q\) to unitary transformations in the GNS (pre-)Hilbert space and of time reversal maps (involutive anti-symplectic diffeomorphisms of \(T^*Q)\) to anti-unitary transformations. We show that the fixed-point set of any involutive time reversal map is either empty or a Lagrangian submanifold. Moreover, we compare our approach to concepts using integral formulas of generalized Moyal-Weyl type. Furthermore we show that the usual WKB expansion with respect to a projectable Lagrangian submanifold can be formulated by a GNS construction. Finally we prove that any homogeneous star product on any cotangent bundle is strongly closed, i.e., the integral over \(T^*Q\) w.r.t. the symplectic volume vanishes on star-commutators. An alternative Fedosov type deduction of the star product of standard ordered type using a deformation of the algebra of symmetric contravariant tensor fields is given.

### MSC:

53D55 | Deformation quantization, star products |

81S10 | Geometry and quantization, symplectic methods |

37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

### Keywords:

deformation quantization; cotangent bundle; homogeneous star products of Weyl ordered type; differential operator representations; formal GNS construction; fixed-point set; Lagrangian submanifold; WKB expansion
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\textit{M. Bordemann} et al., J. Geom. Phys. 29, No. 3, 199--234 (1999; Zbl 0989.53060)

### References:

[1] | Abraham, R; Marsden, J.E, Foundations of mechanics, (1985), Addison-Wesley Reading, MA |

[2] | Arnal, D; Cortet, J.C; Molin, P; Pinczon, G, Covariance and geometrical invariance in quantization, J. math. phys., 24, 2, 276-283, (1983) · Zbl 0515.22015 |

[3] | Bates, S; Weinstein, A, Lectures on the geometry of quantization, Berkeley mathematics lecture notes, vol. 8, (1995) |

[4] | Bayen, F; Flato, M; Fronsdal, C; Lichnerowicz, A; Sternheimer, D; Bayen, F; Flato, M; Fronsdal, C; Lichnerowicz, A; Sternheimer, D, Deformation theory and quantization, Ann. phys., Ann. phys., 111, 111-151, (1978), part II · Zbl 0377.53025 |

[5] | Bertelson, M; Cahen, M; Gutt, S, Equivalence of star products, Université libre de bruxelles, travaux de mathématiques, fascicule, 1, 1-15, (1996) |

[6] | Bordemann, M, On the deformation quantization of super-Poisson brackets, (May 1996), q-alg/9605038 |

[7] | Bordemann, M; Brischle, M; Emmrich, C; Waldmann, S, Subalgebras with converging star products in deformation quantization: an algebraic construction for \(CP\^{}\{n\}\), J. math. phys., 37, 12, 6311-6323, (1996) · Zbl 0923.58024 |

[8] | Bordemann, M; Neumaier, N; Waldmann, S, Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. math. phys., 198, 363-396, (1998) · Zbl 0968.53056 |

[9] | Bordemann, M; Waldmann, S, Formal GNS construction and states in deformation quantization, Comm. math. phys., 195, 549-583, (1998) · Zbl 0989.53057 |

[10] | Bordemann, M; Waldmann, S, Formal GNS construction and WKB expansion in deformation quantization, (), 315-319 · Zbl 1166.53321 |

[11] | Cahen, M; Gutt, S; Rawnsley, J, Quantization of Kähler manifolds. II, Trans. am. math. soc., 337, 73-98, (1993) · Zbl 0788.53062 |

[12] | Connes, A; Flato, M; Sternheimer, D, Closed star products and cyclic cohomology, Lett. math. phys., 24, 1-12, (1992) · Zbl 0767.55005 |

[13] | DeWilde, M; Lecomte, P.B.A, Star-products on cotangent bundles, Lett. math. phys., 7, 235-241, (1983) · Zbl 0514.53031 |

[14] | DeWilde, M; Lecomte, P.B.A, Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. math. phys., 7, 487-496, (1983) · Zbl 0526.58023 |

[15] | DeWilde, M; Lecomte, P.B.A, Formal deformations of the Poisson Lie algebra of a symplectic manifold and star products. existence, equivalence, derivations, () |

[16] | Emmrich, C, Equivalence of extrinsic and intrinsic quantization for observables not preserving the vertical polarization, Comm. math. phys., 151, 515-530, (1993) · Zbl 0767.58015 |

[17] | Emmrich, C, Equivalence of Dirac and intrinsic quantization for non-free group actions, Comm. math. phys., 151, 531-542, (1993) · Zbl 0767.58016 |

[18] | Fedosov, B, A simple geometrical construction of deformation quantization, J. diff. geom., 40, 213-238, (1994) · Zbl 0812.53034 |

[19] | Fedosov, B, Deformation quantization and index theory, (1996), Akademie Verlag Berlin · Zbl 0867.58061 |

[20] | Gerstenhaber, M; Schack, S, Algebraic cohomology and deformation theory, () · Zbl 0676.16022 |

[21] | Helgason, S, Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press New York · Zbl 0451.53038 |

[22] | Kontsevich, M, Deformation quantization of Poisson manifolds, (September 1997), q-alg/9709040 |

[23] | Nest, R; Tsygan, B, Algebraic index theorem, Comm. math. phys., 172, 223-262, (1995) · Zbl 0887.58050 |

[24] | Nest, R; Tsygan, B, Algebraic index theorem for families, Adv. math., 113, 151-205, (1995) · Zbl 0837.58029 |

[25] | Pflaum, M.J, Local analysis of deformation quantization, () · Zbl 0848.58030 |

[26] | Pflaum, M.J, The normal symbol on Riemannian manifolds, New York J. math., 4, 97-125, (1998) · Zbl 0903.35099 |

[27] | Pflaum, M.J, A deformation theoretical approach to Weyl quantization on Riemannian manifolds, Lett. math. phys., 45, 277-294, (1998) · Zbl 0995.53057 |

[28] | Ruiz, J.M, The basic theory of power series, (1993), Vieweg-Verlag Braunschweig |

[29] | Underhill, J, Quantization on a manifold with connection, J. math. phys., 19, 9, 1932-1935, (1978) · Zbl 0426.58011 |

[30] | Widom, H, A complete symbolic calculus for pseudodifferential operators, Bull. sc. math., 104, 19-63, (1980) · Zbl 0434.35092 |

[31] | Woodhouse, N, Geometric quantization, (1980), Clarendon Press Oxford · Zbl 0458.58003 |

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