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Triply Special Relativity is a deformation of Special Relativity based on three fundamental parameters, that describes a noncommutative geometry on a curved spacetime, preserving the Lorentz invariance and the principle of relativity. Its symmetries are generated by a 14-parameter nonlinear algebra. In this paper, we discuss a generalization of the original model and construct its realizations on a canonical phase space. We also investigate in more detail its classical limit, obtained by replacing the commutators by Poisson brackets.

We discuss the generalized Yang Poisson models. We construct generalizations of the Yang Poisson algebra related to $\mathfrak{o}(1,5)$ algebra discussed by Meljanac and Mignemi (2023). The exact realizations of this generalized algebra on canonical phase space are presented and the corresponding differential equations are solved in simple cases. Furthermore, we discuss the Poisson algebras related to $\mathfrak{o}(3,3)$ and $\mathfrak{o}(2,4)$ algebras.

We propose the doubly $\kappa$-dependent Yang quantum phase space which describes the generalization of $D = 4$ Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed $\kappa$-Snyder model. Our model of $D=4$ relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: $(M,R)$ specifying the standard Yang model and $(\kappa,\tilde{\kappa})$ characterizing the Born-dual $\kappa$-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter $\rho$ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of $D=4$ Yang model and the new quantum Yang models described algebraically by quantum-deformed $\hat{o}(1,5)$ algebras., Comment: 10 pages; published version

We discuss here how, when higher-order effects in the parameter $\frac{\ell}{\hbar} |\vec{p}|$, related to the lattice spacing $\ell$, are considered, pristine graphene, and other Dirac materials, can be used as tabletop systems where generalized commutation relations are naturally realized. Such generalized algebras of quantization, which lead to generalized versions of the Heisenberg uncertainty principle, are under intense scrutiny these days, as they could manifest a fundamental length scale of spacetime. Despite the efforts and the many intriguing results, there are no experimental signatures of any generalized uncertainty principle (GUP). Therefore, our results here, which tell how to use tabletop physical systems to test certain GUPs in analog experiments, should be of interest to practitioners of quantum gravity. We identify three different energy regimes that we call ``layers'', where the physics is still of a Dirac type but within precisely described limits. The higher the energy, the more sensitive the Dirac system becomes to the effects of the lattice. Here such lattice plays the role of a discrete space where the Dirac quasi-particles live. With the goals just illustrated, we had to identify the mapping between the high-energy coordinates, $X^i$, and the low-energy ones, $x^i$, i.e., those measured in the lab. We then obtained three generalized Heisenberg algebras. For two of them we have the noticeable result that $X^i = x^i$, and for the third one we obtained an improvement with respect to an earlier work: the generalized coordinates expressed in terms of the standard phase space variables, $X^i(x,p)$, and higher order terms., Comment: 13 pages, no figures but a nice diagram. To appear in the proceedings of the conference DICE 2022: Spacetime, Matter, Quantum Mechanics, Castiglioncello, Italy, 19-23 September 2022

We propose $\hbar$-expansions as perturbative solutions of quantum extended Snyder and Yang models, with $\hbar$-independent classical zero-th order terms responsible for the spontaneous breaking of $D=4$ and $D=5$ de Sitter symmetries. In such models, with algebraic basis spanned by $\hat o(D,1)$ Lie algebra generators, we relate the vacuum expectation values (VEV) of the spontaneously broken generators with the Abelian set of ten (Snyder, $D=4$) or fifteen (Yang, $D=5$) antisymmetric tensorial generalized coordinates, which are also used as zero order input for obtaining the perturbative solutions of quantum extended Snyder and Yang models. In such a way we will attribute to these Abelian generalized coordinates the physical meaning of the order parameters describing spontaneous symmetry breaking (SSB). It appears that the consecutive terms in $\hbar$-power series can be calculated explicitly if we supplement the SSB order parameters by the dual set of tensorial commutative momenta., Comment: 11 pages; published version

We describe, in an algebraic way, the $\kappa$-deformed extended Snyder models, that depend on three parameters $\beta, \kappa$ and $\lambda$, which in a suitable algebra basis are described by the de Sitter algebras ${o}(1,N)$. The commutation relations of the algebra contain a parameter $\lambda$, which is used for the calculations of perturbative expansions. For such $\kappa$-deformed extended Snyder models we consider the Heisenberg double with dual generalized momenta sector, and provide the respective generalized quantum phase space depending on three parameters mentioned above. Further, we study for these models an alternative Heisenberg double, with the algebra of functions on de Sitter group. In both cases we calculate the formulae for the cross commutation relations between generalized coordinate and momenta sectors, at linear order in $\lambda$. We demonstrate that in the commutators of quantum space-time coordinates and momenta of the quantum-deformed Heisenberg algebra the terms generated by $\kappa$-deformation are dominating over $\beta$-dependent ones for small values of $\lambda$., Comment: 11 pages, no figures, accepted version

We show that graphene, in its simplest form and settings, is a practical table-top realization of the analog of exotic quantum gravity scenarios, which are speculated to lead to certain generalized Heisenberg algebras. In particular, we identify three different energy regimes (the ``layers'') where the physics is still of a pseudorelativistic (Dirac) type but more and more sensitive to the effects of the lattice. This plays here a role analog to that of a discrete space, where the Dirac quasiparticles live. This work improves and pushes further earlier results, where the physical meaning of the high energy momenta was clear, but the conjugate coordinates only had a purely abstract description. Here we find the physical meaning of the latter by identifying the mapping between the high-energy coordinates and low-energy ones, i.e., those measured in the lab. We then obtain two generalized Heisenberg algebras that were not noticed earlier. In these two cases, we have the striking result that the high-energy coordinates just coincide with the standard ones, measured in the lab. A third generalized Heisenberg algebra is obtained, and it is an improvement of the results obtained earlier in two respects: we now have an expression of the generalized coordinates in terms of the standard phase-space variables, and we obtain higher order terms. All mentioned results clearly open the doors to table-top experimental verifications of many generalized uncertainty principle-corrected predictions of the quantum gravity phenomenology., Comment: 32 pages, 3 figures, matches published version

It has been pointed out that different choices of momenta can be associated to the same noncommutative spacetime model. The question of whether these momentum spaces, related by diffeomorphisms, produce the same physical predictions is still debated. In this work, we focus our attention on a few different momentum spaces that can be associated to the Galilean Snyder noncommutative spacetime model and show that they produce different predictions for the energy spectrum of the harmonic oscillator.

We analyze the model of a self-interacting $\phi^4_{\star}$ scalar field theory in Snyder-de Sitter space. After analytically computing the one-loop beta functions {in the small noncommutativity and curvature limit}, we solve numerically the corresponding system of differential equations, showing that in this limit the model possesses at least one regime in which the theory is asymptotically free. Moreover, in a given region of the parameter space we also observe a peculiar running of the parameter associated to the curvature, which changes its sign and therefore can be interpreted as a transition from an IR de-Sitter space to and UV anti-de Sitter one., Comment: 14 pages, 3 figures. v2: corrects factors 2 in eq. (7) and (8), and eq. (19) regarding the discussion of $m_{\rm eff}$ and $\alpha_{\rm eff}$

We show in general that for a relativistic theory with curved momentum space, i.e.~a theory with deformed relativistic symmetries, the physical velocity of particles coincides with their group velocity. This clarifies a long-standing question about the discrepancy between coordinate and group velocity for this kind of theories. The first evidence that this was the case had been obtained at linear order in the deformation parameter in Phys.Lett.B700(2011)150 for the specific case of $\kappa$-momentum space. The proof was based on the recent understanding of how relative locality affects these scenarios. We here rely again on a careful implementation of relative locality effects, and obtain our result for a generic (relativistic) curved momentum space framework at all orders in the deformation/curvature parameter. We also discuss the validity of this result when the deformation depends on the coordinates as well as on the momenta.

We study the effects of relative locality dynamics in the case of the Snyder model. Several properties of this model differ from those of the widely studied $\kappa$-Poincar\'e models: for example, in the Snyder case the action of the Lorentz group is preserved, and the composition law of momenta is deformed by terms quadratic in the inverse Planck energy. From the investigation of time delay and dual curvature lensing we deduce that, because of these differences, in the Snyder case the properties of the detector are essential for the observation of relative locality effects. The deviations from special relativity do not depend on the energy of the particles and are much smaller than in the $\kappa$-Poincar\'e case, so that are beyond the reach of present astrophysical experiments. However, these results have a conceptual interest, because they show that relative-locality effects can occur even if the action of the Lorentz group on phase space is not deformed.

Using a quantization of the nonassociative and noncommutative Snyder phi^4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter beta. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative phi^4 products can generate different momentum-conserving integrals. Finally most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore infrared divergences should exist, in general, in the Snyder phi^4 theory. We consider infrared divergences at the limit p -> 0 as UV-IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative phi^4 products. We also discuss the extrapolation of the Snyder deformation parameter beta to negative values as well as certain general properties of one-loop quantum corrections in Snyder phi^4 theory at the zero-momentum limit., Comment: 14 pages, 3 figures, version to be published in Phys. Rev. D

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 Poincar\'{e} 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., Comment: 15 pages, references added, matches published version

In this article we define and quantize a truncated form of the nonassociative and noncommutative Snyder phi^4 field theory using the functional method in momentum space. More precisely, the action is approximated by expanding up to the linear order in the Snyder deformation parameter beta, producing an effective model on commutative spacetime for the computation of the two-, four- and six-point functions. The two- and four-point functions at one loop have the same structure as at the tree level, with UV divergences faster than in the commutative theory. The same behavior appears in the six-point function, with a logarithmic UV divergence and renders the theory unrenormalizable at beta^1 order except for the special choice of free parameters s_1=-s_2. We expect effects from nonassociativity on the correlation functions at beta^1 order, but these are cancelled due to the average over permutations., Comment: 13 pages, 6 figures. Version to be published in Phys.Rev.D

We consider cosmological inflation generated by a scalar field slowly rolling off from a de Sitter maximum of its potential. The models belong to the class of hilltop models and represent the most general model of this kind in which the scalar potential can be written as the sum of two exponentials. The minimally coupled Einstein-scalar gravity theory obtained in this way is the cosmological version of a two-scale generalization of known holographic models, allowing for solitonic solutions interpolating between an AdS spacetime in the infrared and scaling solutions in the ultraviolet. We then investigate cosmological inflation in the slow-roll approximation. Our model reproduces correctly, for a wide range of its parameters, the most recent experimental data for the power spectrum of primordial perturbations. Moreover, it predicts inflation at energy scales of four to five orders of magnitude below the Planck scale. At the onset of inflation, the mass of the tachyonic excitation, i.e. of the inflaton, turns out to be seven to eight orders of magnitude smaller than the Planck mass., Comment: 11 pages, 4 figures. Matches published version