q-Deformed Poincaré algebra

Abstract

The ^-differential calculus for the g-Minkowski space is developed. The algebra of the ^-derivatives with the g-Lorentz generators is found giving thê -deformation of the. Poincare algebra. The reality structure of the g-Poincare algebra is given. The reality structure of the ^-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for thê -Poincare algebra are obtained making it a Hopf algebra.

On the deformability of Heisenberg algebras

“…Its spectrum then is similar to the one of p in (19). Actually the momentum eigenstates |n π 0 can be realized by ordinary functions.…”

“…(35) shows that the element Λ is well defined in the q-algebra and in the classical algebra as well. Its coproduct is grouplike but not consistent with the involution on the quantum group [19]. Using this element the original and the conjugated derivatives can be related by the formula:∂…”

A q-Lorentz algebra from q-deformed harmonic oscillators

“…The existence and form of a q-deformed Minkowski spacetime have been considered in detail recently by a number of authors [8,10,[20][21][22][23]. Here we shall simply consider how the deformed Minkowski space algebra of Ogievetsky et al can be related to the algebra of the q-SHO.…”

“…In previous discussions [8,10] generators for q-Minkowski space coordinates were built out of q-spinor bilinears, their commutation relations being determined by the SU q (2) qspinor relations. In these discussions the q-spinors were taken as coordinates of the noncommutative quantum plane {(x, y) : xy = qyx}, the underlying carrier space of SU q (2) [24].…”

“…Fundamental representations are obtained and the generators are reformulated into qdeformed rotations and boosts. It is also noted that there exists a mapping from q-boson operators to the q-Minkowski coordinates of Ogievetsky et al [9,10].…”