Double Quantization

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 phasespace 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 λ-Minkwoski and R 3 λ noncommutative spaces.

Comments on noncommutative quantum mechanical systems associated with Lie algebras

“…I thank Vladislav Kupriyanov for important comments, for bringing my attention to Refs. [8,9] and for informing me about the paper [10] prior to publication. I also acknowledge useful discussions with Maxim Kontsevich and Zoran Škoda.…”

Localization and observers in $\varrho$-Minkowski spacetime

“…In [13] a field theory on this space has been built; in the same paper a different physical identification of the non-commuting variables has been also considered, with time a commuting coordinate. The latter has been studied in [14] in the context of Poisson gauge models and in [15] in the context of double quantization 2 .…”

Towards quantum mechanics on the curved cotangent bundle

“…In the context of nonrelativistic quantum mechanics, this idea has already been investigated algebraically, i.e. from the point of view of deformations of the Galilean algebra [31][32][33][34][35].…”