Star products made (somewhat) easier

Abstract

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. By using a polydifferential representation for deformed coordinates ${\hat{x}}^j$ we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.

20 pages, v2, v3: comments and references added

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Cited by in corpus (34)

• Topics in Noncommutative Geometry Inspired Physics
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Position-dependent noncommutativity in quantum mechanics
• Hydrogen atom on curved noncommutative space
• Noncommutativity due to spin
• Noncommutative $R^d$ via closed star product
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Quantum mechanics with coordinate dependent noncommutativity
• Non-commutative deformation of Chern-Simons theory
• Semiclassical Lindblad master equation for spin dynamics
• Poisson gauge theory
• Nonassociative Weyl star products
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• Dynamical noncommutativity
• Gauging the twisted Poincare symmetry as noncommutative theory of gravitation
• $L_{\infty }$-Bootstrap Approach to Non-Commutative Gauge Theories
• Involutive representations of coordinate algebras and quantum spaces
• Equilibrium Chiral Magnetic Effect: spatial inhomogeneity, finite temperature, interactions
• Tensor calculus on noncommutative spaces
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Precise Wigner-Weyl calculus for lattice models
• Weak associativity and deformation quantization
• Feynman Rules in terms of the Wigner transformed Green functions
• On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space
• Gauge field theories with covariant star-product
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Covariant star product on symplectic and Poisson spacetime manifolds
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• Conserved symmetries in noncommutative quantum mechanics
• Poisson electrodynamics on $\kappa$-Minkowski space-time
• Discrete Wigner-Weyl calculus for the finite lattice
• Quantum spaces, central extensions of Lie groups and related quantum field theories
• Non-commutative gauge symmetry from strong homotopy algebras
• On the computation of star products