Double Quantization

TL;DR

This paper develops a mathematical framework combining spacetime and phase-space noncommutativity using Drinfel'd twist, applied to specific noncommutative spaces relevant in quantum gravity.

Contribution

It introduces a novel method to describe simultaneous noncommutativity of spacetime and phase space via Drinfel'd twist, addressing a gap in quantum gravity models.

Findings

Constructed a Drinfel'd twist for phase space noncommutativity.

Applied the method to $ ext{lambda}$-Minkowski and $ ext{R}^3_ ext{lambda}$ spaces.

Provides a unified approach to quantum and spacetime noncommutativity.

Abstract

In a quantum gravity theory, it is expected that the classical notion of spacetime disappears, leading to a quantum structure with new properties. A possible way to take into account these quantum effects is through a noncommutativity of spacetime coordinates. In the literature, there is not a clear way to describe at the same time a noncommutativity of spacetime and the phase-space noncommutativity of quantum mechanics. In this paper we address this issue by constructing a Drinfel'd twist in phase space which deals with both quantizations. This method can be applied to a noncommutativity which involves only space, leaving time aside. We apply our construction to the so-called $\lambda$-Minkwoski and $\mathbb{R}^3_\lambda$ noncommutative spaces.