Abstract
We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.
Highlights
Since the beginning, the research of a quantum field theory brought with it the problem of ultraviolet divergences and it was already Heisenberg who proposed the idea of noncommutative spaces as a possible solution [1]
The result is given by a closed analytical expression
In this paper we have constructed the non-associative star product related to a Hermitian realisation of noncommutative coordinates, originally introduced by Snyder
Summary
The research of a quantum field theory brought with it the problem of ultraviolet divergences and it was already Heisenberg who proposed the idea of noncommutative spaces as a possible solution [1]. In this paper we construct the non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder [2]. This construction is performed using the method proposed in [38,51,55]. The result is given by a closed analytical expression This twist does not satisfy the cocycle condition, but we prove that it reproduces the star product and the coproducts of the momenta and the Lorentz generators. It generates a quasi-bialgebra and quasi-bialgebroid structure related to the Snyder space.
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