Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. Method for general construction of star product is presented. Corresponding twist, expressed in terms of phase space coordinates, in Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincar\'e-Weyl generators or $\mathfrak{gl}(n)$ generators, are constructed and R-matrix is discussed. Classification of linear realizations leading to vector-like deformed phase spaces is given. There are 3 types of spaces: $i)$ commutative spaces, $ii)$ $\kappa$-Minkowski spaces and $iii)$ $\kappa$-Snyder spaces. Corresponding star products are $i)$ associative and commutative (but non-local), $ii)$ associative and non-commutative and $iii)$ non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed.Comment: 20 pages, version accepted for publication in EPJ

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

Solutions of coupled BPS equations for two-family Calogero and matrix models

We consider a large N, two-family Calogero and matrix model in the Hamiltonian, collective-field approach. The Bogomol'nyi limit appears and the solutions to the coupled Bogomol'nyi-Prasad-Sommerfeld equations are given by the static soliton configurations. We find all solutions close to constant and construct exact one-parameter solutions in the strong-weak dual case. Full classification of these solutions is presented.Comment: latex, 15 pages, no figure

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

Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach

We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.Comment: 15 pages,no figures; v2 with completed references, which appeared in PL

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

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

Analytical Results for Trapped Weakly Interacting Bosons in Two Dimensions

We consider a model of N two-dimensional bosons in a harmonic trap with translational and rotational invariant, weak two-particle interaction. We present in configuration space a systematical recursive method for constructing all wave functions with angular momentum L and corresponding energies and apply it to L\leq 6 for all N. The lower and the upper bounds for interaction energy are estimated. We analitically confirm the conjecture of Smith et al. that elementary symmetric polynomial is the ground state for repulsive delta interaction, for all N\geq L up to L\leq 6. Additionally, we find that there exist vanishing-energy solutions for L\geq N(N-1), signalizing the exclusive statistics. Finally, we consider briefly the case of attractive power-like potential r^k, k>-2, and prove that the lowest-energy state is still the one in which all angular momentum is absorbed by the center-of-mass motion.Comment: RevTex, 13 page