Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions＇ Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.     

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions' Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.     

Deformation Quantization with Traces

In this Letter we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a constant volume form and a Poisson bivector field on ^d such that div=0, the Kontsevich star product with the harmonic angle function is cyclic, i.e. (f*g)·h·= (g*h)·f· for any three functions f,g,h on (for which the integrals make sense). We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and an arbitrary volume form, and prove a generalization of the Connes–Flato–Sternheimer conjecture on closed star products in the Poisson case.     

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the Formality conjecture), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (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 (M) and ( (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.     

Weyl quantization and star products

Many non-linear classical mechanical systems arise as the symplectic reductions of linear systems. The star products on the corresponding quantized algebras can be derived from the Weyl-Moyal product on the algebras of the linear systems. An algebraic approach to Berezin quantization is sketched.     

Invariant Star Products on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> and the Canonical Trace

We calculate the canonical trace and use the Fedosov–Nest–Tsygan index theorem to obtain the characteristic class for a star product on S ². We show how, for this simple example, it is possible to extract the relevant information needed to use the Fedosov–Nest–Tsygan index theorem from a local calculation.This revised version was published online in March 2005 with corrections to the cover date.     

Non-Abelian Symplectic Cuts and the Geometric Quantization of Noncompact Manifolds

Let (M, ) be a Hamiltonian U(n)-space with proper moment map. In the case where n = 1, Lerman constructed a one-parameter family of Hamiltonian U(1)-spaces M called the symplectic cuts of M. We generalize this construction to Hamiltonian U(n) spaces. Motivated by recent theorems that show that 'quantization commutes with reduction,' we next give a definition of geometric quantization for noncompact Hamiltonian G-spaces with proper moment map, and use our cutting technique to illustrate the proof of existence of such quantizations in the case of U(n) spaces. We then show (Theorem 1) that such quantizations exist in general.     

Oscillatory Integral Formulae for Left-invariant Star Products on a Class of Lie Groups

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)     

Projectively Equivariant Quantization Map

Let M be a manifold endowed with a symmetric affine connection . The aim of this Letter is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T* M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on M. This map is an isomorphism, for almost all values of certain constants, and it depends only on the projective class of the affine connection .     

Projectively Equivariant Quantization for Differential Operators Acting on Forms

In this Letter, we show the existence of a natural and projectively equivariant quantization map depending on a linear torsion-free connection for the spaces of differential operators mapping p-forms into functions on an arbitrary smooth manifold M. We show how this result implies the existence over of an sl _m+1-equivariant quantization for the spaces .This revised version was published online in March 2005 with corrections to the cover date.     

A Remark on Nonequivalent Star Products via Reduction for CPn

In this Letter, we construct nonequivalent star products on CPⁿ by phase space reduction. It turns out that the nonequivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on Cⁿ⁺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 CPⁿ are obtained by this construction.     

Parametrizing Equivalence Classes of Invariant Star Products

Let (M,,) be a symplectic manifold endowed with a symplectic connection . Let Symp(M,) be the group of symplectic transformations of (M,) and Aff(M,) be the group of affine transformations of the affine manifold (M, ). In this Letter, we show that, for any subgroup G of Symp(M,) Aff(M,), the set of G-equivalence classes of G-invariant star products on (M,) is canonically parametrized by the set of sequences of elements belonging to the second de Rham cohomology space of the G-invariant de Rham complex on M.     

Energy for Two-electron Quantum Dots: The Quantization Rule Approach

The energy spectra for two electrons in a parabolic quantum dot are calculated by the quantization rule approach. The numerical results are in excellent agreement with the results by the method of integrating directly the Schr?dinger equation, and better than those by the WKB method and the WKB-DP method.     

Cohomology of Good Graphs and Kontsevich Linear State Products

Following the ideas presented in q-alg/9709040, we give the definition of Kontsevich star products for linear Poisson structures on ^d. We prove that all these structures are equivalent and can be defined by integral formulae. Finally, we characterize, among these star products, the Gutt and Duflo star products.     

A Fedosov Star Product of the Wick Type for Kähler Manifolds

In this Letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of the Wick type on every Kähler manifold by a straightforward generalization of the corresponding star product in Cn: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure, we give an existence proof of such star products for any Kähler manifold.     

A Gerbe Obstruction to Quantization of Fermions on Odd-Dimensional Manifolds with Boundary

We consider the canonical quantization of fermions on an odd-dimensional manifold with boundary, with respect to a family of elliptic Hermitian boundary conditions for the Dirac Hamiltonian. We show that there is a topological obstruction to a smooth quantization as a function of the boundary conditions. The obstruction is given in terms of a gerbe and its Dixmier–Douady class is evaluated.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

A Remark on Formal KMS States in Deformation Quantization

Within 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[[]]-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.