Quantization by cochain twists and nonassociative differentials

Abstract

We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to $O({\hslash }^3)$, but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociavitity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalisation of differential structures. The quantisations are induced by a classical group covariance and include: enveloping algebras $U_{\hslash }(g)$ as quantisations of $g^{\ast }$, a Fedosov-type quantisation of the sphere $S^2$ under a Lorentz group covariance, the Mackey quantisation of homogeneous spaces, and the standard quantum groups $C_q[G]$. We also consider the differential quantisation of $R^n$ for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.

30 pages ams-latex no figures; version submitted to journal (updated section 5 to stronger form)

References in corpus (6)

• Nonassociative tori and applications to T-duality
• The Dirac operator on SU_q(2)
• Gauge theory on nonassociative spaces
• Noncommutative Model with Spontaneous Time Generation and Planckian bound
• Braided Cyclic Cocycles and Non-Associative Geometry
• Exact Black Hole Solutions in Noncommutative Gravity

Cited by in corpus (15)

• Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics
• Gauge theory on nonassociative spaces
• Noncommutative Model with Spontaneous Time Generation and Planckian bound
• On kappa-deformation and triangular quasibialgebra structure
• Almost commutative Riemannian geometry: wave operators
• Twisting and kappa-Poincare
• Noncommutative Differentials on Poisson-Lie groups and pre-Lie algebras
• Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions
• Noncommutative spherically symmetric spacetimes at semiclassical order
• Fuzzy and discrete black hole models
• Quantum Koszul formula on quantum spacetime
• Triangular quasi-Hopf algebra structures on certain non-semisimple quantum groups
• Topics in Noncommutative Geometry
• Nonassociatuve generalization of supersymmetry
• Poisson Principal Bundles