Noncommutative Spherically Symmetric Spaces

TL;DR

This paper explores noncommutative spherically symmetric spaces in three dimensions, extending Snyder's model to include dilation generators and constructing a compatible Laplacian with a creation-annihilation operator framework.

Contribution

It introduces a generalized noncommutative space incorporating dilation operators and develops a spherically symmetric Laplacian with a suitable spectrum.

Findings

Constructed a noncommutative Laplacian with correct spectral properties

Realized the algebra using creation and annihilation operators

Proposed a truncation approach for the Hilbert space

Abstract

We examine some noncommutative spherically symmetric spaces in three space dimensions. A generalization of Snyder's noncommutative (Euclidean) space allows the inclusion of the generator of dilations into the defining algebra of the coordinate and rotation operators. We then construct a spherically symmetric noncommutative Laplacian on this space having the correct limiting spectrum. This is presented via a creation and annihilation operator realization of the algebra, which may lend itself to a truncation of the Hilbert space.