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 $T^{\ast }Q$ by means of the Fedosov procedure using a symplectic torsion-free connection on $T^{\ast }Q$ 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 $T^{\ast }Q$ polynomial in the momenta (where an arbitrary fixed torsion-free connection ${\nabla }_0$ on $Q$ is used). Motivated by the flat case $T^{\ast }{R}^n$ 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.

31 pages, LaTeX2e, no pictures

Cited by in corpus (53)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Deformation quantization in the teaching of quantum mechanics
• On higher spin interactions with matter
• The characteristic classes of Morita equivalent star products on symplectic manifolds
• Variations on deformation quantization
• Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
• Star products made (somewhat) easier
• Universality of Fedosov's Construction for Star Products of Wick Type on Pseudo-Kähler Manilfolds
• Wigner Measures in Noncommutative Quantum Mechanics
• Natural star products on symplectic manifolds and quantum moment maps
• Quantum Magnetic Algebra and Magnetic Curvature
• Noncommutative geometry, gauge theory and renormalization
• Sur l'existence d'une prescription d'ordre naturelle projectivement invariante
• States and representations in deformation quantization
• Affine Quantization on the Half Line
• A vertical exterior derivative in multisymplectic geometry and a graded Poisson bracket for nontrivial geometries
• (Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions
• D-branes on Group Manifolds and Deformation Quantization
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• Traces for star products on the dual of a Lie algebra
• Cotangent bundle quantization: Entangling of metric and magnetic field
• Symplectic Dirac-Kähler Fields
• Deformation Quantization of Poisson Structures Associated to Lie Algebroids
• A Remark on Formal KMS States in Deformation Quantization
• Wick Quantization of Cotangent Bundles over Riemannian Manifolds
• Convergence of the Gutt Star Product
• Gravitational mass-shift effect in the Standard Model
• Hermitian star products are completely positive deformations
• Deformation Quantization of Hermitian Vector Bundles
• Completely positive inner products and strong Morita equivalence
• Fluctuations as stochastic deformation
• A Remark on the Deformation of GNS Representations of *-Algebras
• Pseudodifferential operators on manifolds with linearization
• Convergence of Star Product: From Examples to a General Framework
• H-äquivariante Morita-Äquivalenz und Deformationsquantisierung
• Beyond Coherent State Quantization
• Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer
• Classification of polarized deformation quantizations
• On the relation between geometric and deformation quantization
• Morita equivalence bimodules for Wick type star products
• Geometric and deformation quantization
• Symplectic Connections of Ricci Type and Star Products
• Quantum mechanics and classical trajectories
• Convergence of the Wick Star Product
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product
• Bohr-Sommerfeld Star Products
• Deformation Quantization in Singular Spaces
• Quantization via Deformation of Prequantization
• Deformation quantization on the cotangent bundle of a Lie group
• Deformation Quantization for Supermanifolds via Gelfand-Kazhdan Descent
• Stargenfunctions, generally parametrized systems and a causal formulation of phase space quantum mechanics
• Involutions and Representations for Reduced Quantum Algebras
• Lifting central invariants of quantized Hamiltonian actions