Scalar field theory on noncommutative Snyder spacetime

Abstract

We construct a scalar field theory on the Snyder noncommutative space-time. The symmetry underlying the Snyder geometry is deformed at the co-algebraic level only, while its Poincaré algebra is undeformed. The Lorentz sector is undeformed at both the algebraic and co-algebraic level, but the coproduct for momenta (defining the star product) is non-coassociative. The Snyder-deformed Poincaré group is described by a non-coassociative Hopf algebra. The definition of the interacting theory in terms of a nonassociative star product is thus questionable. We avoid the nonassociativity by the use of a space-time picture based on the concept of the realization of a noncommutative geometry. The two main results we obtain are (i) the generic (namely, for any realization) construction of the co-algebraic sector underlying the Snyder geometry and (ii) the definition of a nonambiguous self-interacting scalar field theory on this space-time. The first-order correction terms of the corresponding Lagrangian are explicitly computed. The possibility to derive Noether charges for the Snyder space-time is also discussed.

Nonassociative Snyder ϕ4 quantum field theory

“…On the other hand, Snyder's spacetime [28], the subject of this investigation, belongs to a rather different type of models [29][30][31][32], and is defined by the phase space commutation relations,…”

“…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.…”

UV-IR mixing in nonassociative Snyder ϕ4 theory

“…1 Functions Ψðβp 2 Þ and Φ μν ðpÞ are constrained so that the Jacobi identities hold. Detailed computations and discussions of the Snyder realizations are given in previous works [35,37]. The Snyder model has also been treated from different points of view in [39][40][41][42][43].…”

“…Snyder spacetime [32], the subject of the present investigation, belongs to a class of models that have been introduced and investigated using the Hopf-algebra formalism in [34][35][36][37][38]. These generalizations can be studied in terms of noncommutative coordinatesx μ and momentum generators p μ , that span a deformed Heisenberg algebra [37] …”

Snyder-type space–times, twisted Poincaré algebra and addition of momenta

“…The Hopf algebra associated with the Snyder model has been studied in a series of papers [6,7,8], where the model has been generalised and the star product, coproduct and antipodes have been calculated using the method of realisations. A different approach was used in [9], where the Snyder model was considered in a geometrical perspective as a coset in momentum space, and results equivalent to those of refs.…”