A dynamical 2-dimensional fuzzy space

Abstract

The noncommutative extension of a dynamical 2-dimensional space-time is given and some of its properties discussed. Wick rotation to euclidean signature yields a surface which has as commutative limit the doughnut but in a singular limit in which the radius of the hole tends to zero.

Gravity and the structure of noncommutative algebras

“…The essential ingredients in the definition of the map are the Leibniz rules and the assumption (2.1) on the structure of the differential calculus. Although there have been found [16,17,4,6] numerous particular examples, there is not yet a systematic discussion of either the range or kernel of the map. We have here to a certain extent alleviated this, but only in the context of perturbation theory around a vacuum and even then, only in the case of a high-frequency wave.…”

On noncommutative spherically symmetric spaces

“…We have here an indication of the importance of the calculi in the description of the geometries: the differential calculus usually introduced on a space described by the Heisenberg algebra is flat, while here the subspace (r, τ ) has a constant negative curvature, [12]. One can find even an example of an algebra over which there are two different calculi with geometries having as the commutative limit two different topologies.…”

Non Commutative Quantum Mechanics in Time - Dependent Backgrounds

“…So if we focus on point one, that is ∆q 1 → ǫ > 0 then the second point becomes fuzzier as time passes, that is ∆q 2 → ∞ . We can say that the background space is a dynamical two -dimensional fuzzy space (21). We will transform the problem of the non commuting two dimensional space, to a problem of two coupled harmonic oscillators in a more familiar two dimensional quantum mechanical space.…”