Nikolai Neumaier

Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation

Abstract:In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T*Q by means of the Fedosov procedure using a symplectic torsion-free connection on T*Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T*Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T*ℝn another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.

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

Abstract 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 [M. Bordemann, N. Neumaier, S. Waldmann, Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. Math. Phys. 198 (1998) 363–396]) we construct differential operator representations via the formal GNS construction (see [M. Bordemann, S. Waldmann, Formal GNS construction and states in deformation quantization, Comm. Math. Phys. 195 (1998) 549–583]). The positive linear functional is integration over Q with respect to some fixed density and is shown to yield a reasonable version of the Schrodinger 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.

On representations of star product algebras over cotangent spaces on Hermitian line bundles

Abstract For every formal power series B of closed two-forms on a manifold Q and every value of an ordering parameter κ∈[0,1] we construct a concrete star product ★κB on the cotangent bundle T ∗ Q . The star product ★κB is associated to the symplectic form on T ∗ Q given by the sum of the canonical symplectic form ω and the pull back of B to T ∗ Q . Delignes characteristic class of ★κB is calculated and shown to coincide with the formal de Rham cohomology class of π ∗ B divided by iλ. Therefore, every star product on T ∗ Q corresponding to the canonical Poisson bracket is equivalent to some ★κB. It turns out that every ★κB is strongly closed. In this paper, we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on Q. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion.

Homology of formal deformations of proper étale Lie groupoids

Abstract In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a non-commutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

Universality of Fedosov's construction for star products of Wick type on pseudo-Kähler manifolds

In this paper we construct star products on a pseudo-Kahler manifold (M, ω, I) using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on Cn. Having fixed the used connection to be the pseudo-Kahler connection these star products shall depend on certain data given by a formal series of closed two-forms on M and a certain formal series of symmetric contravariant tensor fields on M. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Delignes characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and with respect to anti-holomorphic directions in the second argument. These star products are in fact strongly related relvtar products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type (1, 1) on M. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-Kahler manifold. Due to this result we can make some interesting observations concerning these star products; we can show that all these star products are of Vey type and in addition we can uniquely characterize the ones that have the complex conjugation incorporated as an anti-automorphism.

Complete positivity of Rieffel's deformation quantization by actions of R d

In this paper we consider C*-algebraic deformations a la Rieffel and show that every state of the undeformed algebra can be deformed into a state of the deformed algebra in the sense of a continuous field of states. The construction is explicit and involves a convolution operator with a particular Gauss function.

A C * -Algebraic Model for Locally Noncommutative Spacetimes

Locally noncommutative spacetimes provide a refined notion of noncommutative spacetimes where the noncommutativity is present only for small distances. Here we discuss a non-perturbative approach based on Rieffel’s strict deformation quantization. To this end, we extend the usual C*-algebraic results to a pro-C*-algebraic framework.

Deformation quantization of surjective submersions and principal fibre bundles

Abstract In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In the case of a principal fibre bundle we additionally require invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all di¤erential operators on the total space is computed and gives a deformation of the algebra of vertical di¤erential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E , where E ! M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevichs formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and self- equivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E the integration with respect to a density with vanishing modular vector field defines a trace functional.

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence, and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.