Twist for Snyder space

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.

Nonassociative Snyder ϕ4 quantum field theory

“…with Σ 1 being defined in (36). The sum of three Σ 1 's contains also contributions from two additional diagrams obtained from the diagram in Fig.…”

“…The Snyder model has been studied in a series of papers [30][31][32][33][34][35][36] and the associated Hopf algebra investigated in [30] and [36], where the model has been generalized and the star product, coproducts and antipodes have been calculated using the method of realizations. A different approach was used in [35], where the Snyder model was considered in a geometrical perspective as a coset in momentum space, and the results are equivalent to those of Refs.…”

“…[31,32]. A further generalization of Snyder spacetime deformations was recently introduced in [36][37][38]. Also several nonassociative star/cross product geometries and related quantum field theories have been discussed recently in [39].…”

Associative realizations of the extended Snyder model

“…We recall here some of the most relevant recent advances in Snyder theory: in [9] the Snyder algebra was generalized in such a way to maintain the Lorentz invariance; in [6] the coproduct was calculated, in [7] the same problem was investigated from a geometrical point of view, using the fact that the momentum space of Snyder can be identified with a coset space; the twist was investigated in [10,11]. The construction of a field theory was first addressed in [6,7] and then examined in more detail in [12].…”

Snyder-de Sitter Meets the Grosse-Wulkenhaar Model

“…On the other hand, two of the most interesting solutions to the Lorentz noninvariance problem are the deformation (or twisting) of the symmetry algebra [10] (in the so called Moyal plane) or the consideration of the original Snyder space. In this regard several developments have been achieved in the last years -among others, the behaviour of particles in Snyder space [11,12,13], the star product, the coproduct and antipodes of its Hopf algebra, the twist operator [14,15,16,17], and a scalar QFT defined on it [18,19,20] have been studied.…”