Snyder-de Sitter Meets the Grosse-Wulkenhaar Model

Abstract

We study an interacting λ φ 4 ⋆ scalar field defined on Snyder-de Sitter space. Due to the noncommutativity as well as the curvature of this space, the renormalization of the two-point function differs from the commutative case. In particular, we show that the theory in the limit of small curvature and noncommutativity is described by a model similar to the Grosse-Wulkenhaar one. Moreover, very much akin to what happens in the Grosse-Wulkenhaar model, our computation demonstrates that there exists a fixed point in the renormalization group flow of the harmonic and mass terms.

Asymptotic freedom for $$\lambda \phi ^4_{\star }$$ QFT in Snyder–de Sitter space

“…It is therefore naturally of interest to further investigate the origin of the harmonic term. It has been observed that the introduction of such term may be justified by the assumption of a de Sitter spacetime background [12]. For this reason, we extend our previous study of the Snyder scalar QFT to the case of a de Sitter background.…”

Quantum Mechanics of the Extended Snyder Model

“…For example, it is clear from (4) that the uncertainty relations for ∆x i ∆p j in the Weyl realization depend on the expectation values of the tensorial degrees of freedom, while in the classical realization (5) they coincide with those of the standard Snyder model. Also, modifications to the Casimir force between conducting plates could be evaluated on the lines of the calculations performed in [26] for the standard case.…”

“…and the interaction term becomes V = Mω 2 8 ∑ ij x i p j x i p j − x j p i + x ik p jk x il p jl − x jl p il + 2x i p j x ik p jk − x jk p ik +β −2 x j p ij x k p ik + β 2 x ij p j x ik p k − x i p j p jk x ik − p j x i x ik p jk (26) Defining, for β = 0,…”

The beauty of curved momentum space

“…After several years of an almost total oblivion, these ideas regained vigor with the advent of the 𝑞-deformed Poincaré algebra [12], obtained as a limit of the Drinfeld-Jimbo deformation of the anti-de Sitter algebra so(3, 2). Among the subsequent works devoted to the study of particles and fields in 𝜅-Minkowski and Snyder spaces, we can mention [35,[56][57][58][59][60][61][62][63][64][65][66][67][68] and especially those dealing with fermions [69][70][71][72][73].…”