Noncommutative spacetime realized in Ad Sn+1 space: Nonlocal field theory out of noncommutative spacetime

S Naka,E Umezawa,H Toyoda,T Takanashi

doi:10.1093/ptep/ptu022

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.

Highlights

The κ-Minkowski spacetime is a noncommutative spacetime characterized by an algebraic structure with a constant κ other than the light velocity; in this sense, the framework of κ-Minkowski spacetime is called as doubly special relativity (DSR)[1]
This framework is symmetric under the κ-Poincare transformation, which reduces to the Lorentz transformation according as κ → ∞
The algebraic structure of Snyder’s spacetime is slightly different from that of the κ Minkowski spacetime, the symmetry under the Lorentz boost is broken in this spacetime too

Summary

Introduction

The κ-Minkowski spacetime is a noncommutative spacetime characterized by an algebraic structure with a constant κ other than the light velocity; in this sense, the framework of κ-Minkowski spacetime is called as doubly special relativity (DSR)[1]. The algebraic structure of Snyder’s spacetime is slightly different from that of the κ Minkowski spacetime, the symmetry under the Lorentz boost is broken in this spacetime too In both types of noncommutative space times, the dispersion relation of particles embedded in this spacetime becomes highly non-linear due to κ = 0. The invariant wave equation under the κ-Poincare transformations is, P4(k)Ψ = 0 or P (k)μP (k)μΨ = 0 We may read those equations as non-local field equations in the Minkowski spacetime by substitution kμ.

Non-local field in the background of noncommutative spacetime

An attempt of interacting fields

Summary and discussion

B The boost in terms of wave number vectors