arXiv subjects

M. Lagraa

Publications and source records attributed to M. Lagraa.

10 recordsLinked to original sources

arXiv2018

On the Hamiltonian formalism of the Tetrad-Gravity with fermions

We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero. Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.

gr-qc↗

arXiv2006

On the ADHM construction of noncommutative U(2) k-instanton

The basic objects of the ADHM construction are reformulated in terms of elements of the $A_θ(R^4)$ algebra of the noncommutative $R_θ^4$ space. This new formulation of the ADHM construction makes possible the explicit calculus of the U(2) instanton number which is shown to be the product of a trace of finite rank projector of the Fock representation space of the algebra $A_θ(R^4)$ times a noncommutative version of the winding number.

hep-th↗

arXiv2006

The flux of noncommutative U(1) instanton through the fuzzy spheres

From the ADHM construction on noncommutative $R_θ^4$ we investigate different U(1) instanton solutions tied by isometry trasformations. These solutions present a form of vector fields in noncommutative $R_θ^3$ vector space which makes possible the calculus of their fluxes through fuzzy spheres. We establish the noncommutative analog of Gauss theorem from which we show that the flux of the U(1) instantons through fuzzy spheres does not depend on the radius of these spheres and it is invariant under isometry transformations.

hep-th↗

arXiv2002

Coherent State Induced Star-Product on $R^3_λ$ and the Fuzzy Sphere

Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, $R^2_θ\times R^2_θ$, we obtain a star-product for the noncommutative (fuzzy) $R^3_λ$ defined by $[x^i,x^j]=iλε_{ijk}x^k$. Furthermore, we show that there is a projection function which allows us to reduce the functions on $R^3_λ$ to that of the fuzzy sphere, and hence we introduce a new star-product on the fuzzy sphere. We will then briefly discuss how using our method one can extract information about the field theory on fuzzy sphere and $\rrlam$ from the corresponding field theories on $R_{theta}\times R_θ$ space.

hep-th↗

arXiv2001

The Quantum Spheres and their Embedding into Quantum Minkowski Space-Time

We recast the Podles spheres in the noncommutative physics context by showing that they can be regarded as slices along the time coordinate of the different regions of the quantum Minkowski space-time. The investigation of the transformations of the quantum sphere states under the left coaction of the ${\cal SO}_{q}(3)$ group leads to a decomposition of the transformed Hilbert space states in terms of orthogonal subspaces exhibiting the periodicity of the quantum sphere states.

hep-th↗

arXiv2001

The Boosts in the Noncommutative Special Relativity

From the quantum analog of the Iwasawa decomposition of $SL(2,C)$ group and the correspondence between quantum $SL(2,C)$ and Lorentz groups we deduce the different properties of the Hopf algebra representing the boost of particles in noncommutative special relativity. The representation of the boost in the Hilbert space states is investigated and the addition rules of the velocities are established from the coaction. The q-deformed Clebsch-Gordon coefficients descibing the transformed states of the evolution of particles in noncommutative special relativity are introduced and their explicit calculation are given.

math-ph↗

arXiv2001

Measurability of observables in the noncommutative special relativity

We adapt the axioms of the quantum mechanics to the quantum Minkowski space-time coordinates and their transformations under the quantum Lorentz group to show how we can formulate the noncommutative special relativity and its quantum physical observables. we establish in this formalism the quantum analog of the lifetime dilatation formula and the relativistic relations between the energy-momentum four-vector and the mass and the velocity. From the explicit construction of states, we establish the causality principle in the noncommutative special relativity and show that for a free particle moving in the quantum Minkowski space-time, only the length of the velocity and one of its components can be measured exactly and simultaneously. In addition these observables present discret spectrums which imply quantized lifetimes of moving unstable particles.

math-ph↗

arXiv1999

On the Quantum Lorentz Group

The quantum analogues of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indiceswe construct a quantum Minkowsky metric. In this framework, we show explicitely the correspondance between the SL(2,C) and Lorentz quantum groups.

math.QA↗

arXiv1997

The Noncommutative Inhomogeneous Hopf Algebra

From the bicovariant first order differential calculus on inhomogeneous Hopf algebra ${\cal B}$ we construct the set of right-invariant Maurer-Cartan one-forms considered as a right-invariant basis of a bicovariant ${\cal B}$-bimodule over which we develop the Woronowicz's general theory of differential calculus on quantum groups. In this formalism, we introduce suitable functionals on ${\cal B}$ which control the inhomogeneous commutation rules. In particular we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of ${\cal B}$ and translations are controled by different R-matrices satisfying nontrivial characteristic equations.

q-alg↗

arXiv1997

Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra

We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.

q-alg↗