κ-deformed spacetime from twist

Jong-Geon Bu,Hyeong-Chan Kim,Youngone Lee,Chang Hyon Vac,Jae Hyung Yee

doi:10.1016/j.physletb.2008.03.058

Abstract

We twist the Hopf algebra of igl(n,R) to obtain the κ-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The κ-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional κ-Minkowski spacetime, but its Hopf algebra structure is different from the well-known κ-deformed Poincaré algebra in that it has larger symmetry algebra than the κ-Minkowski case. There are some physical models which consider this symmetry [R. Percacci, Phys. Lett. B 144 (1984) 37; R. Percacci, Geometry of Nonlinear Field Theories, World Scientific, Singapore, 1986; L. Smolin, Nucl. Phys. B 132 (1978) 138; R. Floreanini, R. Percacci, Class. Quantum Grav. 7 (1990) 975]. Incidentally, we obtain the canonical (θ-deformed) non-commutative spacetime from canonically twisted igl(n,R) Hopf algebra.

Highlights

There have been extensive efforts to understand the gravity and quantum physics in a unified viewpoint
If one finds a twist that gives κ-deformed commutation relation between coordinates from Poincare Hopf algebra, it would be very useful in constructing quantum field theory in κ-deformed spacetime since we can use the irreducible representations of Poincare algebra for the κ-noncommutative quantum field theory
In this paper we focus on twisting the Hopf algebra of igl(n, R) as a symmetry algebra

Summary

INTRODUCTION

There have been extensive efforts to understand the gravity and quantum physics in a unified viewpoint. Chaichian et al [18] use the twist deformation of quantum group theory to interpret the symmetry of the canonical noncommutative field theory as twisted Poincare symmetry. If one finds a twist that gives κ-deformed commutation relation (especially time-like κdeformed noncommutativity) between coordinates from Poincare Hopf algebra, it would be very useful in constructing quantum field theory in κ-deformed spacetime since we can use the irreducible representations of Poincare algebra for the κ-noncommutative quantum field theory. There has been some attempts to obtain the κ-commutation relation of the coordinate system, Eq (2), from twisting the Poincare Hopf algebra [38],[39],[40]. As far as we know, there is no twist of the Poincare algebra which gives a time-like κ-deformed coordinate commutation relation.

BRIEF SUMMARY OF TWISTING HOPF ALGEBRA

DISCUSSION