Abstract
The star product usually associated to the Snyder model of noncommutative geometry is nonassociative, and this property prevents the construction of a proper Hopf algebra. It is however possible to introduce a well-defined Hopf algebra by including the Lorentz generators and their conjugate momenta into the algebra. In this paper, we study the realizations of this extended Snyder spacetime, and obtain the coproduct and twist and the associative star product in a Weyl-ordered realization, to first order in the noncommutativity parameter. We then extend our results to the most general realizations of the extended Snyder spacetime, always up to first order.
Highlights
Since the origin of quantum field theory there have been proposals to add a new scale of length to the theory in order to solve the problems connected to ultraviolet divergences
The star product usually associated to the Snyder model of noncommutative geometry is nonassociative, and this property prevents the construction of a proper Hopf algebra
This was the first example of a noncommutative geometry: the length scale should enter the theory through the commutators of spacetime coordinates, see [3,4]
Summary
Since the origin of quantum field theory there have been proposals to add a new scale of length to the theory in order to solve the problems connected to ultraviolet divergences. A way to reconcile the discreteness of spacetime with Lorentz invariance was originally proposed by Snyder [2] a long time ago This was the first example of a noncommutative geometry: the length scale should enter the theory through the commutators of spacetime coordinates, see [3,4]. We consider a Weyl realization of the algebra in terms of the extended Heisenberg algebra, and generalize it to the most general one compatible with Lorentz invariance at order β, including the one obtained in [7], and compute the coproduct and the star product in the general case.
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