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 $O(\hbar^3)$, 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 $U_\hbar(g)$ as quantisations of $g^*$, 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.