Noncommutative Spacetime Realized in $AdS_{n+1}$ Space

TL;DR

This paper constructs a noncommutative spacetime model within $AdS_{n+1}$ space, exploring its algebraic structure and invariant wave equations, aiming to connect noncommutative geometry with finite quantum field theory amplitudes.

Contribution

It introduces a novel noncommutative spacetime framework based on $AdS_{n+1}$ space with SO(2,n) symmetry, extending previous models in de Sitter space.

Findings

Realization of noncommutative spacetime in $AdS_{n+1}$ space.

Development of an invariant non-local wave equation.

Potential for finite loop amplitude calculations.

Abstract

In $\kappa$-Minkowski spacetime, the coordinates are Lie algebraic elements such that time and space coordinates do not commute, whereas space coordinates commute each other. The non-commutativity is proportional to a Planck-length-scale constant $\kappa^{-1}$, which is a universal constant other than the light velocity under the $\kappa$-Poincare transformation. In this sense, the spacetime has a structure called as "Doubly Special Relativity". Such a noncommutative structure is known to be realized by SO(1,4) generators in 4-dimensional de Sitter space. In this paper, we try to construct a nonommutative spacetime having commutative n-dimensional Minkowski spacetime based on $AdS_{n+1}$ space with SO(2,n) symmetry. We also study an invariant wave equation corresponding to the first Casimir invariant of this symmetry as a non-local field equation expected to yield finite loop amplitudes.