Noncommutative field theory on $\mathbb{R}^3_\lambda$

TL;DR

This paper explores a noncommutative deformation of three-dimensional space, introducing a new Laplacian operator and analyzing its implications for field theory on the fuzzy sphere foliation.

Contribution

It constructs a natural matrix basis for $R^3_l$ and proposes a novel Laplacian operator not of Jacobi type, advancing noncommutative field theory models.

Findings

Defined a matrix basis adapted to $R^3_l$

Proposed a new Laplacian operator for the deformed algebra

Discussed implications for noncommutative field theory

Abstract

We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a foliation of $\mathbb{R}^3$ into fuzzy spheres. We first review the construction of a natural matrix basis adapted to $\mathbb{R}^3_\lambda$. We thus consider the problem of defining a new Laplacian operator for the deformed algebra. We propose an operator which is not of Jacobi type. The implication for field theory of the new Laplacian is briefly discussed.