The DFR-Algebra for Poisson Vector Bundles

TL;DR

This paper constructs a broad family of $C^*$-algebras extending the DFR quantum space-time algebra, using a fiberwise quantization approach for Poisson vector bundles and manifolds, unifying several quantum space models.

Contribution

It introduces a new $C^*$-algebra framework that generalizes the DFR model to Poisson vector bundles and manifolds, enabling fiberwise quantization.

Findings

Includes the original DFR-model as a special case

Provides a fiberwise quantization method for general Poisson manifolds

Constructs $C^*$-algebras as modules over $C_0(M)"

Abstract

The aim of the present paper is to present the construction of a general family of $C^*$-algebras that includes, as a special case, the "quantum space-time algebra" first introduced by Doplicher, Fredenhagen and Roberts. To this end, we first review, within the $C^*$-algebra context, the Weyl-Moyal quantization procedure on a fixed Poisson vector space (a vector space equipped with a given bivector, which may be degenerate). We then show how to extend this construction to a Poisson vector bundle over a general manifold $M$, giving rise to a $C^*$-algebra which is also a module over $C_0(M)$. Apart from including the original DFR-model, this method yields a "fiberwise quantization" of general Poisson manifolds.