arXiv subjects

Stefan Waldmann

Publications and source records attributed to Stefan Waldmann.

69 records · Page 4Linked to original sources

arXiv2001

The characteristic classes of Morita equivalent star products on symplectic manifolds

In this paper we give a complete characterization of Morita equivalent star products on symplectic manifolds in terms of their characteristic classes: two star products $\star$ and $\star'$ on $(M,ω)$ are Morita equivalent if and only if there exists a symplectomorphism $ψ:M\longrightarrow M$ such that the relative class $t(\star,ψ^*(\star'))$ is $2 π\im$-integral. For star products on cotangent bundles, we show that this integrality condition is related to Dirac's quantization condition for magnetic charges.

math.QA↗

arXiv2001

On the Representation Theory of Deformation Quantization

In this contribution to the proceedings of the 68eme Rencontre entre Physiciens Theoriciens et Mathematiciens on Deformation Quantization I shall report on some recent joint work with Henrique Bursztyn on the representation theory of *-algebras arising from deformation quantization as I presented this in my talk.

math.QA↗

arXiv2000

A Remark on the Deformation of GNS Representations of *-Algebras

Motivated by deformation quantization we investigate the algebraic GNS construction of *-representations of deformed *-algebras over ordered rings and compute their classical limit. The question if a GNS representation can be deformed leads to the deformation of positive linear functionals. Various physical examples from deformation quantization like the Bargmann-Fock and the Schr{ö}dinger representation as well as KMS functionals are discussed.

math.QA↗

arXiv2000

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper $^*$-algebras $\mathcal A$ over rings $\ring C = \ring{R}(i)$, where $\ring R$ is an ordered ring and $i^2 = -1$, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) $\mathcal A$-valued inner product. For $A=C^\infty(M)$, M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star-product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of $C^\infty(M)$ and $Γ^\infty(\End(E))$ and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of $C^*$-algebras. We also discuss the semi-classical geometry arising from these deformations.

math.QA↗

arXiv2000

Ideals and Formal Morita Equivalence of Algebras

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita $^*$-equivalence of rings with involution, and strong Morita equivalence of $C^*$-algebras. Second, in the general setting of $^*$-algebras over $\ring C$, we define `closed' $^*$-ideals as the ones occuring as kernels of $^*$-representations of these algebras on pre-Hilbert spaces. These ideals form a lattice which we show is invariant under formal Morita equivalence. This result, when applied to Pedersen ideals of $C^*$-algebras, recovers the so-called Rieffel correspondence theorem. The triviality of the minimal element in the lattice of closed ideals, called the `minimal ideal', is also a formal Morita invariant and this fact can be used to describe a large class of examples of $^*$-algebras over $\ring C$ with equivalent representation theory but which are not formally Morita equivalent. We finally compute the closed $^*$-ideals of some $^*$-algebras arising in differential geometry.

math.QA↗

arXiv2000

Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the *-representation theory of such *-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of *-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two *-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate *-representations of the two *-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over *-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.

math.QA↗

arXiv1999

On Positive Deformations of *-Algebras

Motivated by deformation quantization we consider $^*$-algebras over ordered rings and their deformations: we investigate formal associative deformations compatible with the $^*$-involution and discuss a cohomological description in terms of a Hermitian Hochschild cohomology. As an ordered ring allows for a meaningful definition of positive functionals and as the formal power series with coefficients in an ordered ring are again an ordered ring we define a deformation to be positive if any positive linear functional of the undeformed algebra can be deformed into a positive linear functional of the deformed algebra. We discuss various examples and prove in particular that star products on symplectic manifolds are positive deformations.

math.QA↗

arXiv1999

BRST Cohomology and Phase Space Reduction in Deformation Quantisation

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a `quantum' Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is `sufficiently nice', e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counter examples are discussed.

math.QA↗

arXiv1999

Locality in GNS Representations of Deformation Quantization

In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over $\mathbb C[[λ]]$ and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish `thermal' from `pure' representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schrödinger representation on cotangent bundles. Finally we prove a formal version of the Tomita-Takesaki theorem.

math.QA↗

arXiv1998

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

For every formal power series $B=B_0 + λB_1 + O(λ^2)$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $κ\in [0,1]$ we construct a concrete star product $\star^B_κ$ on the cotangent bundle $π: T^*Q\to Q$. The star product $\star^B_κ$ is associated to the formal symplectic form on $T^*Q$ given by the sum of the canonical symplectic form $ω$ and the pull-back of $B$ to $T^*Q$. Deligne's characteristic class of $\star^B_κ$ is calculated and shown to coincide with the formal de Rham cohomology class of $π^*B$ divided by $\imλ$. Therefore, every star product on $T^*Q$ corresponding to the Poisson bracket induced by the symplectic form $ω+ π^*B_0$ is equivalent to some $\star^B_kappa$. It turns out that every $\star^B_kappa$ 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.

math.QA↗

arXiv1998

A Remark on Non-equivalent Star Products via Reduction for CP^n

In this paper we construct non-equivalent star products on CP^n by phase space reduction. It turns out that the non-equivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on C^{n+1} {0} into a quantum momentum map and the corresponding momentum value into a quantum momentum value such that the level set, i.~e. the `constraint surface', of the quantum momentum map coincides with the classical one. All equivalence classes of star products on CP^n are obtained by this construction.

math.QA↗

arXiv1998

A Remark on Formal KMS States in Deformation Quantization

In the framework of deformation quantization we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[λ]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.

math.QA↗

arXiv1997

Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications

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 q-alg/9707030) we construct differential operator representations via the formal GNS construction (see q-alg/9607019). 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 Lagrangean 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 Lagrangean 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.

q-alg↗

arXiv1997

Formal GNS Construction and States in Deformation Quantization

In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex $C^*$-algebras: in this approach the corresponding positive linear functionals (`states') take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations as the Bargmann and Schrödinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of the positive functional) a formal solution to the time-dependent Schrödinger equation is shown to exist. Moreover, we show that for every Kähler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.

q-alg↗

arXiv1997

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

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 $\nabla_0$ on $Q$ is used). Motivated by the flat case $T^* R^n$ 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.

q-alg↗