Homogeneous Fedosov star products on cotangent bundles. II: GNS representations, the WKB expansion, traces, and applications.

*(English)*Zbl 0989.53060Summary: 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##### 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.