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Stjepan Meljanac

Publications and source records attributed to Stjepan Meljanac.

At least 73 records · Page 4Linked to original sources
arXiv2011

Differential structure on kappa-Minkowski space, and kappa-Poincare algebra

We construct realizations of the generators of the $κ$-Minkowski space and $κ$-Poincaré algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $κ$-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $κ$-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 $κ$-Minkowski space.

math-ph↗
arXiv2011

Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra

We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the Hopf structure and star product are found. Special cases, particularly Snyder and $κ$-Minkowski in Maggiore-type realizations are discussed. The same generalized Hopf algebraic structures are as well considered in the case of an arbitrary allowable kind of realisation and results are given perturbatively up to second order in deformation parameters.

math-ph↗
arXiv2011

Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance

We investigate the properties of kappa-Minkowski spacetime by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg-Weyl algebra. The deformed algebra consists of kappa-Poincare algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an adventages of this approach to consistently construct a star product which has a property that under integration sign it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal, but not also for kappa-Minkowski spacetime. This star product also has generalized trace and cyclic properties and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and by requiring it to be hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachionic modes and basicaly of the very same form. The issue of Lorentz invariance of the theory is also discussed.

hep-th↗
arXiv2010

Scalar Field Theory on Non-commutative Snyder Space-Time

We construct a scalar field theory on the Snyder non-commutative space-time. The symmetry underlying the Snyder geometry is deformed at the co-algebraic level only, while its Poincaré algebra is undeformed. The Lorentz sector is undeformed at both algebraic and co-algebraic level, but the co-product for momenta (defining the star-product) is non-co-associative. The Snyder-deformed Poincaré group is described by a non-co-associative Hopf algebra. The definition of the interacting theory in terms of a non-associative star-product is thus questionable. We avoid the non-associativity by the use of a space-time picture based on the concept of realization of a non-commutative geometry. The two main results we obtain are: (i) the generic (namely for any realization) construction of the co-algebraic sector underlying the Snyder geometry and (ii) the definition of a non-ambiguous self interacting scalar field theory on this space-time. The first order correction terms of the corresponding Lagrangian are explicitly computed. The possibility to derive Noether charges for the Snyder space-time is also discussed.

hep-th↗
arXiv2009

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.

hep-th↗
arXiv2009

Noncommutative Differential Forms on the kappa-deformed Space

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.

hep-th↗
arXiv2008

Gauge transformations and symmetries of integrable systems

We analyze several integrable systems in zero-curvature form within the framework of $SL(2,\R)$ invariant gauge theory. In the Drienfeld-Sokolov gauge we derive a two-parameter family of nonlinear evolution equations which as special cases include the Kortweg-de Vries (KdV) and Harry Dym equations. We find residual gauge transformations which lead to infinintesimal symmetries of this family of equations. For KdV and Harry Dym equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformatinos of Miura type we obtain a sequence of gauge equivalent integrable systems, among them the modified KdV and Calogero KdV equations.

hep-th↗
arXiv2008

Covariant realizations of kappa-deformed space

We study a Lie algebra type $κ$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. Space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.

hep-th↗
arXiv2007

New realizations of Lie algebra kappa-deformed Euclidean space

We study Lie algebra $κ$-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 $κ$-deformed noncommutative space of the Lie algebra type with undeformed Poincar{é} algebra and with the corresponding deformed coalgebra is constructed in a unified way.

hep-th↗
arXiv2007

Leibniz rules for enveloping algebras in symmetric ordering

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.

math.QA↗
arXiv2006

A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

Given a $n$-dimensional Lie algebra $g$ over a field $k \supset \mathbb Q$, together with its vector space basis $X^0_1,..., X^0_n$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i = X^0_i t$ in $g\otimes_k k [[t]]$, where $t$ is a formal variable, as a formal power series in $t$ with coefficients in the Weyl algebra $A_n$. Actually, the theorem is proved for Lie algebras over arbitrary rings $k\supset Q$. We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

math.RT↗
arXiv2006

Calogero Model(s) and Deformed Oscillators

We briefly review some recent results concerning algebraical (oscillator) aspects of the $N$-body single-species and multispecies Calogero models in one dimension. We show how these models emerge from the matrix generalization of the harmonic oscillator Hamiltonian. We make some comments on the solvability of these models.

hep-th↗
arXiv2005

Higgs Masses in the Minimal SUSY SO(10) GUT

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.

hep-ph↗
arXiv2005

SO(10) Group Theory for the Unified Model Building

The complete tables of Clebsh-Gordan (CG) coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given. Explicit expression of all states and corresponding multiplets under standard model gauge group G_{321} = SU(3)_C x SU(2)_L x U(1)_Y, necessary for evaluation of the CG coefficients are presented. The SUSY SO(10) GUT model considered here incudes most of the Higgs irreducible representations usually used in the literature: 10, 45, 54, 120, 126 126-bar and 210. Mass matrices of all G_{321} multiplets are found for the most general superpotential. These results are indispensible for the precision calculations of the gauge coupling unification and proton decay, etc.

hep-ph↗
arXiv2005

Solitons in the Calogero model for distinguishable particles

We consider a large $- N, $ two-family Calogero model in the Hamiltonian, collective-field approach. The Bogomol'nyi limit appears and the corresponding solutions are given by the static-soliton configurations. Solitons from different families are localized at the same place. They behave like a paired hole and lump on the top of the uniform vacuum condensates, depending on the values of the coupling strengths. When the number of particles in the first family is much larger than that of the second family, the hole solution goes to the vortex profile already found in the one-family Calogero model.

hep-th↗
arXiv2004

Lepton Flavour Violation in the Minimal SO(10) GUT Model and in the Standard Model with additional Heavy Dirac Neutrinos

Lepton flavour violation is considered in 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. The models are briefly explained. The structure of the amplitudes for neutrinoless decays of the lepton into a lepton 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.

hep-ph↗
arXiv2004

Detailed Analysis of Proton Decay Rate in the Minimal Supersymmetric SO(10) Model

We consider the minimal supersymmetric SO(10) model, where only one {\bf 10} and one $\bar{\bf 126}$ 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. We analyze the proton decay rate for the dominant decay modes $p \to K^{+} \barν$ by including as many free parameters as possible and varying them. There are two free parameters in the Yukawa sector, while five 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 $1/\tan^2 β$ and the allowed region eventually disappears as $\tan β$ becomes large.

hep-ph↗