Nonassociative Weyl star products

Abstract

Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires (weak) hermiticity and the Weyl condition. We also propose an iterative procedure that allows to compute the star product up to any desired order.

discussion extended, tipos corrected, published version

References in corpus (6)

• New realizations of Lie algebra kappa-deformed Euclidean space
• T-duality and closed string non-commutative (doubled) geometry
• Star products made (somewhat) easier
• Noncommutative $R^d$ via closed star product
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Cited by in corpus (16)

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• Symplectic realisation of electric charge in fields of monopole distributions
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• UV/IR Mixing in Nonassociative Snyder phi^4 Theory
• Nonassociative black holes in R-flux deformed phase spaces and relativistic models of G. Perelman thermodynamics
• Magnetic monopoles and nonassociative deformations of quantum theory
• Monopole star products are non-alternative
• Twist for Snyder space
• Nearly associative deformation quantization
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• Non-commutative gauge symmetry from strong homotopy algebras
• Lagrangian formulation for electric charge in a magnetic monopole distribution