Abstract
In a recent paper, we have studied associative realizations of the noncommutative extended Snyder model, obtained by including the Lorentz generators (tensorial coordinates) and their conjugated momenta. In this paper, we extend this result to also incorporate a covariant realization of the κ-Poincaré spacetime. We obtain the coproduct, the associative star product and the twist in a Weyl-ordered realization, to first order in the noncommutativity parameters. This could help the construction of a quantum field theory based on this geometry.
Highlights
In the last decades, noncommutative geometry has become an important topic both in mathematics and in theoretical physics, where it is considered a promising candidate for describing the structure of spacetime at the Planck scale
The associative star product and the twist in a Weyl-ordered realization, to first order in the noncommutativity parameters. This could help the construction of a quantum field theory based on this geometry
Noncommutative geometry has become an important topic both in mathematics and in theoretical physics, where it is considered a promising candidate for describing the structure of spacetime at the Planck scale
Summary
Noncommutative geometry has become an important topic both in mathematics and in theoretical physics, where it is considered a promising candidate for describing the structure of spacetime at the Planck scale. Poincare model [5] with its associated κ-Minkowski spacetime [6] Both of them are based on a deformation of the commutation relations of the position operators xi: for Snyder space one has∗. [9] it was proposed a formalism for constructing κ-deformations of orthogonal groups by considering transformations that leave invariant generic metric tensors We apply this formalism to the extended Snyder model of ref. We get the extended unified κ-Minkowski Snyder spacetime, with commutation relations [Xi, Xj] = iλ(aiXj − ajXi + βXij), [Xij, Xk] = iλ(ηikXj − ηjkXi − aiXjk + ajXik), [Xij, Xkl] = iλ(ηikXjl − ηilXjk − ηjkXil + ηjlXik) In this way we obtain an unification of the Snyder and κ-Poincare spacetimes using the formalism of extended coordinates Xμν.
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