Twisted $h$ -spacetimes and invariant equations

Abstract

We analyze the h-deformations of the Lorentz group and their associated spacetimes. We prove that they have a twisted character and give explicitly the twisting matrices. After studying the representations of one of the deformed spacetime algebras, we discuss the Klein-Gordon operator. It is found that the h-deformed d'Alembertian has plane wave solutions of the same form as the standard ones. We also give explicit expressions for the h-gamma matrices defining the associated Dirac equations.

Finite Field Theory on Noncommutative Geometries

“…Here µ is a parameter in the operator ∆ with the dimension of mass. In general I(kµ 2 ) is an operator acting on D. In all the examples we shall consider however the space is homogeneous and it reduces to a constant. We can write then j| G(q µ ; q ν′ ) | ī = I(kµ 2 ) j | ī .…”

“…It was postulated some time ago [38,39] that a noncommutative structure at small length scales could introduce an effective cut-off in field theory similar to a lattice but at the same time maintain Lorentz invariance. Recently there has been a revival of this idea and several new examples [27,12,20,2,23,24] have been studied. Models [15,22,21,5] in '1-dimension' have also added to our understanding of the 'lattice' structure.…”

The Dirac operator and gamma matrices for quantum Minkowski spaces

“…Recently, when the present paper was essentially completed, some gamma matrices (satisfying a condition like the condition 2. of Theorem 2.2) appeared also in [2]. That paper contains explicit formulae for the gamma matrices and metric tensor in some cases.…”

“…The gamma matrices (given by (2.25) and (2.9)) and relations among them (the condition 2. of Theorem 2.2 and (2.7)) were announced in [9]. Recently, when the present paper was essentially completed, some gamma matrices (satisfying a condition like the condition 2. of Theorem 2.2) appeared also in [2]. That paper contains explicit formulae for the gamma matrices and metric tensor in some cases.…”

Noncommutative Geometry for Pedestrians

“…It might be argued that since we have made space-time 'noncommutative' we ought to do the same with the Poincaré group. This logic leads naturally to the notion of a q-deformed Poincaré (or Lorentz) group which act on a very particular noncommutative version of Minkowski space called q-Minkowski space [141,142,28,10,30]. The idea of a q-deformation goes back to Sylvester [200].…”