Quantum field theory in generalised Snyder spaces

Abstract

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT

Nonassociative Snyder ϕ4 quantum field theory

“…Note that, in principle, an integral over any star product of two fields under the Hermitian realization condition would reduce to the integral of the usual multiplication. This is certainly true for Moyal and κ-Minkowski cases; however, for the above conjecture in the general case of Snyder spaces, we only have a rigorous proof up to the O(β 2 ) and in the Snyder realization of the full theory [38]. Note also that using Eqs (6)- (8) it is straightforward to show that the 3-cyclicity for nonassociative star products…”

“…[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].…”

UV-IR mixing in nonassociative Snyder ϕ4 theory

“…The star product for the Hermitian Snyder realization is given by [36,37] e ikx ⋆ e iqx ¼ e iDðk;qÞx e iGðk;qÞ ; ð4Þ with the following exact expressions for D μ ðk; qÞ and Gðk; qÞ:…”

“…These generalizations can be studied in terms of noncommutative coordinatesx μ and momentum generators p μ , that span a deformed Heisenberg algebra [37] …”

“…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].…”