BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Geometric quantization of the BRST charge

In the first half of this paper (Sects. 1–4) we generalise the standard geometric quantization procedure to symplectic supermanifolds. In the second half (Sects. 5, 6) we apply this to two examples that exhibit classical BRST symmetry, i.e., we quantize the BRST charge and the ghost number. More precisely, in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with Ad^*-equivariant moment map; in the second example we consider a foliated configuration space, whose cotangent bundle admits the construction of a BRST charge associated to this foliation. We show that the classical BRST symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space, and that geometric quantization gives us a super-unitary representation of this supergroup. Finally we point out how these results are related to reduction at the quantum level, as compared with the reduction at the classical level.Research supported by the Dutch Organization for Scientific Research (NWO)     

A Gauge Invariant Description for the General Conic Constrained Particle from the FJBW Iteration Algorithm

We consider a second-degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting concerning its quantum counterpart with subtleties in its symplectic structure and symmetries. We start with a second-class version of the general conic constrained particle, which encompasses previous versions of circular and elliptical paths discussed in the literature. By applying the symplectic FJBW iteration program, we proceed on to show how a gauge invariant version for the model can be achieved from the originally second-class system. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the essential aspects of the constraint structure. While in the standard Dirac-Bergmann approach there are four second-class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process, we construct a gauge invariant model exhibiting explicitly its BRST symmetry. We obtain the quantum BRST charge and write the Green functions generator for the gauge invariant version. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor, clearly showing that this last one constitutes a particular case of a broader class of theories.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

Getzler symbol calculus and deformation quantization

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler’s pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.     

Local ν-Euler Derivations and Deligne's¶Characteristic Class of Fedosov Star Products¶and Star Products of Special Type

In this paper we explicitly construct local ν-Euler derivations , where the ξ_α are local, conformally symplectic vector fields and the are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by , where is a formal series of closed two-forms on M the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshé Flato Received: 28 June 1999 / Accepted: 11 April 2002?Published online: 11 September 2002     

Spin state tomography

A scheme for measuring the quantum state for an arbitrary spin is proposed that is analogous to the symplectic tomography scheme used to measure quantum states associated with continuous observables such as position and momentum. An invariant form for the spin state density operator is derived in terms of an integral, over the angles which specify the quantization axis, of a product of the measured probability of the values of the spin along a chosen direction and spherical harmonics summed with Clebsch-Gordan functions. Zh. éksp. Teor. Fiz. 112, 796–804 (September 1997)     

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.     

On quantization of polynomial momentum observables

The paper is devoted to quantization of polynomial momentum observables in the cotangent bundle of a smooth manifold. A quantization procedure is proposed allowing to quantize a wide class of functions which are polynomials of any order in the momenta. As an application of the proposed approach, quantum mechanics of a scalar particle in curved space-time is studied.     

Calcul d'un Invariant de Star-Produit Fermé sur une Variété Symplectique

Let M be a symplectic manifold over $ℝ. In [CFS] the authors construct an invariant ϕ in the cyclic cohomology of M for any closed star-product. They compute this invariant in the de Rham complex of M when M=T ^* V. We generalize this result by computing the image of ϕ in the de Rham complex for any symplectic manifold and any star-product and we show how this invariant is related to the general classification of Kontsevich. The proof uses the Riemann–Roch theorem for periodic cyclic chains of Nest–Tsygan.

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Calcul d'un Invariant de Star-Produit Fermé sur une Variété Symplectique

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Received: 30 November 1998 / Accepted: 15 February 1999     

Quantum Dynamical Yang–Baxter Equation¶Over a Nonabelian Base

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra ?=?⊕?, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix naturally corresponds to a Poisson manifold ?^⋆×G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang–Baxter equation (or Gervais–Neveu–Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed. Received: 19 May 2001 / Accepted: 19 November 2001     

Generators of KMS Symmetric Markov Semigroups on {\mathcal{B}({\rm h})} Symmetry and Quantum Detailed Balance

We find the structure of generators of norm-continuous quantum Markov semigroups on B(h){\mathcal{B}({\rm h})} that are symmetric with respect to the scalar product tr (ρ ^1/2 x*ρ ^1/2 y) induced by a faithful normal invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Manifestly fermionic gauge covariant quantization of a massive superparticle

By introducing harmonic variables and additional fermionic coordinates all the fermionic constraints of the massive D=9 superparticle have been converted into first-class constraints. Since all the constraints are irreducible and first class, the BFV-BRST quantization scheme is fully applicable. The BRST charge is found to be of rank one.     

λ→0 limit of 2+1 quantum gravity for arbitrary genus

The abstract quantum algebra of observables for 2+1 gravity is analysed in the limit of small cosmological constant. The algebra splits into two sets with an explicit phase space representation; one set consists of 6g–6commuting elements which form a basis for an algebraic manifold defined by the trace and rank identities; the other set consists of 6g–6 tangent vectors to this manifold. The action of the quantum mapping class group leaves the algebra and algebraic manifold invariant. The previously presented representation forg=2 is analysed in this limit and reduced to a very simple form. The symplectic form forg=2 is computed.     

Symplectic geometry and geometric quantization on supermanifold withU numbers

The usual theory of supermanifold is extended to the case that contains (anti)commuting variable pairs with oppositeU numbers. The symplectic geometry and geometric quantization on such a special manifold are discussed in detail. As applications, the BRST system with finite dimensional first class constraints and bosonic strings are investigated systematically.     

BRST Extension of Geometric Quantization

Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathematically rigorous BRST quantization of the constrained system whose cohomology at ghost number zero recovers the constrained quantum states. Moreover this space of constrained states has a well-defined Hilbert space structure inherited from that of the original system. Treatments of these ideas in the physics literature are more general but suffer from having states with infinite or zero "norms" and thus are not admissible as states. Also BRST operators for many systems require regularization to be well-defined. In our more restricted context, we show that our treatment does not suffer from any of these difficulties.     

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct, we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.