Quantum field theory in generalised Snyder spaces

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 peturbatively 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.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1608.0620

κ\kappa-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations

We consider $\kappa$-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincar\'e algebra is unaltered, while the other is a general extension where the Poincar\'e algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincar\'e-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincar\'e-Weyl algebra.Comment: Title changed and minor changes in the tex

The Energy Operator for a Model with a Multiparametric Infinite Statistics

In this paper we consider energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: $a_{i}a_{j}^{\dagger}-q_{ij}a_{j}^{\dagger}a_{i} = \delta_{ij}, i,j\in I$ with $(q_{ij})_{i,j\in I}$ any hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see Theorem 1) we prove that the number operators are given, with respect to a basis formed by "generalized Lie elements", by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture of two of the authors (S.M and A.P), stated in [8]). These Gram matrices are hermitian generalizations of the Varchenko's matrices, associated to a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes (see [12]). The solution of the inversion problem of such matrices in [9] (Theorem 2.2.17), leads to an effective formula for the number operators studied in this paper. The one parameter case, in the monomial basis, was studied by Zagier [15], Stanciu [11] and M{\o}ller [6].Comment: 24 pages. accepted in J. Phys. A. Math. Ge

Classical dynamics on curved Snyder space

We study the classical dynamics of a particle in nonrelativistic Snyder-de Sitter space. We show that for spherically symmetric systems, parametrizing the solutions in terms of an auxiliary time variable, which is a function only of the physical time and of the energy and angular momentum of the particles, one can reduce the problem to the equivalent one in classical mechanics. We also discuss a relativistic extension of these results, and a generalization to the case in which the algebra is realized in flat space.Comment: 12 pages, LaTeX, version published on CQ

Generalized Poincare algebras, Hopf algebras and kappa-Minkowski spacetime

We propose a generalized description for the kappa-Poincare-Hopf algebra as a symmetry quantum group of underlying kappa-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of kappa-Minkowski algebra realization. For the given realization of kappa-Minkowski spacetime there is a unique kappa-Poincare-Hopf algebra with undeformed Lorentz algebra. We have constructed a three-parameter family of deformed Lorentz generators with kappa-Poincare algebras which are related to kappa-Poincare-Hopf algebra with undeformed Lorentz algebra. Known bases of kappa-Poincare-Hopf algebra are obtained as special cases. Also deformation of igl(4) Hopf algebra compatible with the kappa-Minkowski spacetime is presented. Some physical applications are briefly discussed.Comment: 15 pages; journal version; Physics Letters B (2012

Example of q-deformed Field Theory

The non-relativistic Chern-Simons theory with the single-valued anyonic field is proposed as an example of q-deformed field theory. The corresponding q-deformed algebra interpolating between bosons and fermions,both in position and momentum spaces, is analyzed.A possible generalization to a space with an arbitrary dimension is suggested.Comment: 13 pages,LaTe

New realizations of Lie algebra kappa-deformed Euclidean space

We study Lie algebra $\kappa$-deformed Euclidean space with undeformed rotation algebra $SO_a(n)$ and commuting vectorlike derivatives. Infinitely many realizations in terms of commuting coordinates are constructed and a corresponding star product is found for each of them. The $\kappa$-deformed noncommutative space of the Lie algebra type with undeformed Poincar{\'e} algebra and with the corresponding deformed coalgebra is constructed in a unified way.Comment: 30 pages, Latex, accepted for publication in Eur.Phys.J.C, some typos correcte