Search

We present the exact realization of the extended Snyder model. Using similarity transformations we construct realizations of the original Snyder and the extended Snyder models. Finally, we present the exact new realization of the kappa deformed extended Snyder model., 12 pages

We review deformed quantum phase spaces and their realizations in terms of undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces and Snyder type spaces are considered. Examples of linear realizations of the $\kappa$-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of Yang and triply special relativity models are presented.

Unified graded differential algebra, generated by $\kappa$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $\kappa$-Poincar\'e-Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in \cite{sitarz}. Different realizations of our algebra in terms of super-Heisenberg algebra are presented. For light-like deformations we get 4D bicovariant calculus, with $\kappa$-Poincar\'e-Hopf algebra and present the corresponding twist, which is written in a new covariant way, using Poincar\'e generators only. In the time- and space-like case this twist leads to $\kappa$-Snyder space. Our results might lead to applications in NC quantum field theories (especially electrodynamics and gauge theories), quantum gravity models, and Planck scale physics., Comment: 15 pages, new title, minor changes

We consider different realizations for the momentum sector of kappa-Poincare Hopf algebra, which is associated with a curved momentum space. We show that the notion of the particle mass as introduced recently by Amelino-Camelia et al. in the context of relative-locality is realization independent for a wide class of realizations, up to linear order in deformation parameter l. On the other hand, the time delay formula clearly shows a dependence on the choice of realization.

We consider different realizations for the momentum sector of kappa-Poincare Hopf algebra, which is associated with a curved momentum space. We show that the notion of the particle mass as introduced recently by Amelino-Camelia et al. in the context of relative-locality is realization independent for a wide class of realizations, up to linear order in deformation parameter l. On the other hand, the time delay formula clearly shows a dependence on the choice of realization., 12 pages; extended version of the paper published in Phys. Rev. D 87, 125009 (2013)

We construct realizations of the generators of the $\kappa$-Minkowski space and $\kappa$-Poincar\'{e} algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $\kappa$-Poincar\'{e} algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $\kappa$-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the $\kappa$-Minkowski space., Comment: 20 pages. Accepted for publication in International Journal of Modern Physics A

We discuss how space-time noncommutativity affects the symmetry groups and particle statistics. Assuming that statistics is superselected under a symmetry transformation, we argue that the corresponding flip operator must be twisted. It is argued that the twisted statistics naturally leads to a deformed oscillator algebra for scalar fields in such a background.

Given a finite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules, quantized ``star'' product. We first illuminate a number of interrelations between these constructions and then proceed to study a special case in certain precise sense corresponding to the symmetric or Weyl ordering. This case has been known earlier to be related to computations with Hausdorff series, for example the expression for the star product is in such terms. For the deformed Leibniz rule, hence a coproduct, we present here a new nonsymmetric expression, which is then expanded into a sum of expressions labelled by a class of planar trees, and for a given tree evaluated by Feynman-like rules. These expressions are filtered by a bidegree and we show recursion formulas for the sums of expressions of a given bidegree, and compare the recursions to recursions for Hausdorff series, including the comparison of initial conditions. This way we show a direct corespondence between the Hausdorff series and the expression for twisted coproduct.

We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.

We consider the issue of statistics for identical particles or fields in kappa-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription., Latex file, 8 pages

We investigate a Lie algebra-type $ \kappa$-deformed Minkowski space-time with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of $ \kappa$-Minkowski space. The coproduct and the star product corresponding to each of them are found. Utilizing the properties of the {; ; \em{; ; natural}; ; }; ; realization, we construct a scalar field theory on $ \kappa$-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space.

Given a finite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules, quantized ``star'' product. We first illuminate a number of interrelations between these constructions and then proceed to study a special case in certain precise sense corresponding to the symmetric or Weyl ordering. This case has been known earlier to be related to computations with Hausdorff series, for example the expression for the star product is in such terms. For the deformed Leibniz rule, hence a coproduct, we present here a new nonsymmetric expression, which is then expanded into a sum of expressions labelled by a class of planar trees, and for a given tree evaluated by Feynman-like rules. These expressions are filtered by a bidegree and we show recursion formulas for the sums of expressions of a given bidegree, and compare the recursions to recursions for Hausdorff series, including the comparison of initial conditions. This way we show a direct corespondence between the Hausdorff series and the expression for twisted coproduct., v2: changed title (previous title: Coproduct for symmetric ordering), improved exposition, now uses elsart.cls

We study the collective field formulation of a restricted form of the multispecies Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well- known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.

A quantum matrix oscillator model is shown to reproduce the quantum Hall states in a direct way and is intimately related to the regularized matrix model introduced by Polychronakos. By transferring the consideration to the Bargmann representation with the help of a particular similarity transformation, we show that the quantum matrix oscillator describes the quantum mechanics of electrons in the lowest Landau level with the ground state described by the Laughlin-type wave function.

Lepton Flavor violation is considered on two models, Standard Model with additional heavy Dirac neutrinos and minimal supersymmetric standard model (MSSM) with SO(10) theoretical frame and minimal SUGRA (mSUGRA) SUSY-breaking mediation mechanism. Two models are briefly explained. The structure of the amplitudes for neutrinoless decays of the lepton into lepton and photon and semileptonic neutrinoless decays of a lepton is given. The method for identification of meson content in the quark currents is exposed. The comparison of the amplitudes is made showing which of the processes have the largest branching ratios. The numerical results for the dominant decay rates are presented.

We study a single matrix oscillator with the quadratic Hamiltonian and deformed commutation relations. It is equivalent to the multispecies Calogero model in one dimension, with inverse-square two-body and three-body interactions. Specially, a new matrix realization of the Calogero model for identical particles have been constructed, without using exchange operators. The critical points at which singular behaviour occurs are briefly discussed. At the end, a connection of a quantum matrix oscillator to a modified matrix Chern-Simons model is established. The quantum mechanics of electrons in the lowest Landau level is shown to be described by the matrix oscillator model with the ground state wave function of the Laughlin type.

We explicitly show that minimal supersymmetric SO(10) Higgs-Higgsino mass matrices evaluated by various groups are mutually consistent and correct. We comment on the corresponding results of other authors. We construct one-to-one mappings of our approach to the approaches of other authors.

We explicitly show that minimal SUSY SO(10) Higgs-Higgsino mass matrices evaluated by various groups are mutually consistent and correct. We comment on the corresponding results of other authors. We construct one-to-one mappings of our approach to the approaches of other authors., Comment: 4 pages, revtex4, Changes in the paper : We agree with Ref. hep-ph/0501025 that there there should exist equivalence mappings between results of Higgs mass spectra calculated by three groups of authors. (in three different approaches. However, in the Note added of the Ref. hep-ph/0405074 v3 the wrong mapping was presented. The explicit construction of equivalence mappings to other approches is given. Minor changes in the text. Paper is beeing accepted for publication in Phys. Rev D

We consider the minimal supersymmetric SO(10) model, where only one 10 and one Higgs multiplets have Yukawa couplings with matter multiplets. This model has the high predictive power for the Yukawa coupling matrices consistent with the experimental data of the charged fermion mass matrices, and all the Yukawa coupling matrices are completely determined once a few parameters in the model are fixed. This feature is essential for definite predictions to the proton decay rate through the dimension five operators. Althogh it is not completely general, we analyze the proton decay rate for the dominant decay modes p→ K + by including as many free parameters as possible and varying them. There are two free parameters in the Yukawa sector, while three in the Higgsino sector. It is found that an allowed region exists when the free parameters in the Higgs sector are tuned so as to cancel the proton decay amplitude. The resultant proton lifetime is proportional to 1/tan 2β and the allowed region eventually disappears as tan β becomes large.

The rational Calogero model is one of the most famous and exaustively studied examples of integrable systems. Since its inception more than thirty years ago [1], the model and its various descendants continue to be of interest for both physics and mathematics community. The model describes $N$ identical (single-species) particles on the line which interact through an inverse-square two-body interaction and are subjected to a common confining harmonic force. The inverse-square potential can be regarded as a pure statistical interaction and the model maps to an ideal gas of particles obeying fractional Haldane statistics [2]. However, in Haldane's formulation of statistics there is a possibility of having particles of different species with a mutual statistical coupling parameter depending on the species coupled. This suggests the generalization of the single-species Calogero model to the multispecies Calogero model. Distinguishabillity of the species can be introduced by allowing particles to have different masses and different couplings to each other. While the single-species Calogero model is completely solvable, very little is known about spectra and wave functions of the multispecies Calogero model. Recently, we used an operator method to analyse a one-dimensional multispecies Calogero model with two- and three-body interactions [3]. We succeeded in finding a class of, but not all, exact eigenstates and eigenenergies of the model Hamiltonian. The analysis relied heavily on the SU(1, 1) algebraic structure of the Hamiltonian and once more stressed the importance of the conformal symmetry of the quantum singular oscillator [4]. The universal $SU(1, 1)$ structure of the model permitted us to discuss and partially solve the problem of $F$ interacting families of Calogero type-particles in one dimension [5]. By imposing the conditions for the absence of the three-body interaction, we found certain relations between the coupling constants, sometimes termed as "weak-strong duality".\\ We were also able to generalize the model to arbitrary dimensions [6]. In spite of the complications caused by inevitable appearance of the three-body interaction, we have found ground state and some of the excited states describing the global collective modes.\\ We emphasize that our algebraic analysis is applicable to all systems with underlying $SU(1, 1)$ algebra.

A one-dimensional model with F interacting families of Calogero-type particles is studied. The model includes harmonic, two-body and three-body interactions. We emphasize the universal SU(1, 1) structure of the model. We show how SU(1, 1) generators for the whole system are composed of SU(1, 1) generators of arbitrary subsystems. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the F-family model. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants.

In this paper we study a collections of operators $a(k)$ satisfying the "$q_{kl} $-canonical commutation relations" $a(k)a^{+}(l)-q_{kl}a^{+}(l)a(k) =\delta_{kl} $ (corresponding for $q_{kl}=q$ to Greenberg (infinite) statistics, for $q=\pm 1$ to classical Bose and Fermi statistics).We show that $n!\times n!$ matrices $A_{n}(\{q_{kl}\})$ of scalar products of n-particle states is positive definite for all n if $|q_{kl}, Comment: 63 pages, 1 figure

We study an N-body Calogero model in the S_N-symmetric subspace of the positive definite Fock space. We construct a new algebra of S_N-symmetric operators represented on the symmetric Fock space, and find a natural orthogonal basis by mapping the algebra onto the Heisenberg algebra. Our main result is the bosonic realization of nonlinear dynamical symmetry algebra describing the structure of degenerate levels of the Calogero model.

We discuss common algebraic structure of the matrix Chern-Simons model and the Calogero model.