arXiv subjects

R. P. Malik

Publications and source records attributed to R. P. Malik.

At least 127 records · Page 7Linked to original sources

arXiv2005

Nilpotent Symmetries For A Free Relativistic Particle In Augmented Superfield Formalism

In the framework of the augmented superfield formalism, the local, covariant, continuous and off-shell (as well as on-shell) nilpotent (anti-)BRST symmetry transformations are derived for a $(0 + 1)$-dimensional free scalar relativistic particle that provides a prototype physical example for the more general reparametrization invariant string- and gravitational theories. The trajectory (i.e. the world-line) of the free particle, parametrized by a monotonically increasing evolution parameter $τ$, is embedded in a $D$-dimensional flat Minkowski target manifold. This one-dimensional system is considered on a $(1 + 2)$-dimensional supermanifold parametrized by an even element $τ$ and a couple of odd elements ($θ$ and $\barθ$) of a Grassmannian algebra. The horizontality condition and the invariance of the conserved (super)charges on the (super)manifolds play very crucial roles in the above derivations of the nilpotent symmetries. The geometrical interpretations for the nilpotent (anti-)BRST charges are provided in the framework of augmented superfield approach.

hep-th↗

arXiv2005

Time-Space Noncommutativity And Symmetries For A Massive Relativistic Particle

We show the existence of a time-space noncommutativity (NC) for the physical system of a massive relativistic particle by exploiting the underlying symmetry properties of this system. The space-space NC is eliminated by the consideration of the exact symmetry properties and their consistency with the equations of motion for the above system. The symmetry corresponding to the noncommutative geometry turns out to be the special case of the gauge symmetry such that the mass parameter of the above system becomes noncommutative with the space and time variables. The possible deformations of the gauge algebra between the spacetime variables and the angular momenta are discussed in detail. These modifications owe their origin to the NC of the mass parameter with the space and time variables. The cohomological origin for the above NC is addressed in the language of the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations.

hep-th↗

arXiv2004

Noncommutativity In The Mechanics Of A Free Massless Relativistic Particle

We show the existence of a noncommutative spacetime structure in the context of a complete discussion on the underlying spacetime symmetries for the physical system of a free massless relativistic particle. The above spacetime symmetry transformations are discussed for the first-order Lagrangian of the system where the transformations on the coordinates, velocities and momenta play very important roles. We discuss the dynamics of this system in a systematic manner by exploiting the symplectic structures associated with the four dimensional (non-)commutative cotangent (i.e. momentum phase) space corresponding to a two dimensional (non-)commutative configuration (i.e. target) space. A simple connection of the above noncommutativity (NC) is established with the NC associated with the subject of quantum groups where $SL_{q,q^{-1}} (2)$ transformations play a decisive role.

hep-th↗

arXiv2004

Notoph Gauge Theory as Hodge Theory

In the framework of an extended BRST formalism, it is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form (notoph) gauge theory presents an example of a tractable field theoretical model for the Hodge theory.

hep-th↗

arXiv2004

Geometrical Aspects Of BRST Cohomology In Augmented Superfield Formalism

In the framework of augmented superfield approach, we provide the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST charges and a non-nilpotent bosonic charge. Together, these local and conserved charges turn out to be responsible for a clear and cogent definition of the Hodge decomposition theorem in the quantum Hilbert space of states. The above charges owe their origin to the de Rham cohomological operators of differential geometry which are found to be at the heart of some of the key concepts associated with the interacting gauge theories. For our present review, we choose the two $(1 + 1)$-dimensional (2D) quantum electrodynamics (QED) as a prototype field theoretical model to derive all the nilpotent symmetries for all the fields present in this interacting gauge theory in the framework of augmented superfield formulation and show that this theory is a {\it unique} example of an interacting gauge theory which provides a tractable field theoretical model for the Hodge theory.

hep-th↗

arXiv2004

Jacobi Identity for Poisson Brackets: A Concise Proof

In view of the recent interest in a short proof of the Jacobi identity for the Poisson-brackets, we provide an alternative simple proof for the same. Our derivation is based on the validity of the Leibnitz rule in the context of dynamical evolution.

physics.class-ph↗

arXiv2004

Cohomological Operators and Covariant Quantum Superalgebras

We obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative q-superoscillators for the supersymmetric quantum group GL_{qp} (1|1). In particular, we show that a unique superalgebra, obeyed by the bilinears of fermionic and bosonic noncommutative q-(super)oscillators of GL_{qp} (1|1), is exactly identical to that obeyed by the de Rham cohomological operators. A set of discrete symmetry transformation for a set of GL_{qp} (1|1) covariant superalgebras turns out to be the analogue of the Hodge duality * operation of differential geometry. A connection with an extended BRST algebra obeyed by the nilpotent (anti-)BRST and (anti-)co-BRST charges, the ghost charge and a bosonic charge (which is equal to the anticommutator of (anti-)BRST and (anti-)co-BRST charges) is also established.

hep-th↗

arXiv2004

Wigner's little group and BRST cohomology for one-form Abelian gauge theory

We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of the Wigner's little group for the free one-form Abelian gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well-known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in the framework of Becchi-Rouet-Stora-Tyutin (BRST) cohomology defined in the quantum Hilbert space of states where the Hodge decomposition theorem (HDT) plays a very decisive role.

hep-th↗

arXiv2004

Superfield Approach to Symmetries for Matter Fields in Abelian Gauge Theories

The derivation of the nilpotent (anti-)BRST symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been an long-standing problem in the framework of superfield approach to BRST formalism. These nilpotent (anti-)BRST symmetries for the Dirac fields are derived in the superfield formulation for the interacting Abelian gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. The same type of symmetries are deduced for the 4D complex scalar fields having a gauge invariant interaction with the U(1) gauge field. The above interacting theories are considered on the six $(4 + 2)$-dimensional supermanifold parametrized by four {\it even} spacetime coordinates and a couple of {\it odd} elements of the Grassmann algebra. The invariance of the conserved matter (super)currents and the horizontality condition on the (super)manifolds play very important roles in the above derivations. The geometrical origin and interpretation for all the above off-shell nilpotent symmetries are provided in the framework of superfield formalism.

hep-th↗

arXiv2004

Nilpotent Symmetries for QED in Superfield Formalism

In the framework of superfield approach, we derive the local, covariant, continuous and nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations on the U(1) gauge field $(A_μ)$ and the (anti-)ghost fields $((\bar C)C)$ of the Lagrangian density of the two $(1 + 1)$-dimensional QED by exploiting the (dual-)horizontality conditions defined on the four $(2 + 2)$-dimensional supermanifold. The long-standing problem of the derivation of the above symmetry transformations for the matter (Dirac) fields $(\barψ, ψ)$ in the framework of superfield formulation is resolved by a new set of restrictions on the $(2 + 2)$-dimensional supermanifold. These new physically interesting restrictions on the supermanifold owe their origin to the invariance of conserved currents of the theory. The geometrical interpretation for all the above transformations is provided in the framework of superfield formalism.

hep-th↗

arXiv2003

Superfield Approach to (Non-)local Symmetries for One-Form Abelian Gauge Theory

We exploit the geometrical superfield formalism to derive the local, covariant and continuous Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations and the non-local, non-covariant and continuous dual-BRST symmetry transformations for the free Abelian one-form gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. Our discussion is carried out in the framework of BRST invariant Lagrangian density for the above 4D theory in the Feynman gauge. The geometrical origin and interpretation for the (dual-)BRST charges (and the transformations they generate) are provided in the language of translations of some superfields along the Grassmannian directions of the six ($ 4 + 2)$-dimensional supermanifold parametrized by the four spacetime and two Grassmannian variables.

hep-th↗

arXiv2003

Hamiltonian and Lagrangian Dynamics in a Noncommutative Space

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists {\it equivalently} in the coordinate or the momentum planes embedded in the 4D cotangent manifolds. The signature of this noncommutativity is reflected in the derivation of the first-order Lagrangians where we exploit the most general form of the Legendre transformation defined on the (non-)commutative (co-)tangent manifolds. The second-order Lagrangian, defined on the 4D {\it tangent manifold}, turns out to be the {\it same} irrespective of the noncommutativity present in the 4D cotangent manifolds for the discussion of the Hamiltonian formulation. A connection with the noncommutativity of the dynamics, associated with the quantum groups on the q-deformed 4D cotangent manifolds, is also pointed out.

hep-th↗

arXiv2003

Deformed Traces and Covariant Quantum Algebras for Quantum Groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$

The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator commutation relations are obtained which remain invariant under the coactions of groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$. The $GL_{qp}(2)$ covariant deformed algebra is deduced in terms of the bilinears of bosonic $qp$-oscillators which turns out to be a central extension of the Witten-type deformation of $sl(2)$ algebra. In the case of the supergroup $GL_{qp}(1|1)$, the corresponding covariant algebras contain $N = 2$ supersymmetric quantum mechanical subalgebras.

hep-th↗

arXiv2003

Lagrangian formulation of some q-deformed systems

In the framework of Lagrangian formulation, some q-deformed physical systems are considered. The q-deformed Legendre transformation is obtained for the free motion of a non-relativistic particle on a quantum line. This is subsequently exploited to obtain the Lagrangians for the q-deformed harmonic oscillator and q-deformed relativistic free particle. The Euler-Lagrange equations of motion are derived in a consistent manner with the corresponding Hamilton's equations. The Lagrangian for the q-deformed relativistic particle is endowed with the q-deformed gauge symmetry and reparametrization invariance which are shown to be equivalent only for $ q = \pm 1$.

hep-th↗

arXiv2003

Superfield approach to a novel symmetry for non-Abelian gauge theory

In the framework of superfield formalism, we demonstrate the existence of a new local, covariant, continuous and nilpotent (dual-BRST) symmetry for the BRST invariant Lagrangian density of a self-interacting two ($1 + 1$)-dimensional (2D) non-Abelian gauge theory (having no interaction with matter fields). The local and nilpotent Noether conserved charges corresponding to the above continuous symmetries find their geometrical interpretation as the translation generators along the odd (Grassmannian) directions of the four ($2 + 2)$-dimensional supermanifold.

hep-th↗

arXiv2003

Abelian 2-form gauge theory: special features

It is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form gauge theory provides an example of (i) a class of field theoretical models for the Hodge theory, and (ii) a possible candidate for the quasi-topological field theory (q-TFT). Despite many striking similarities with some of the key topological features of the two $(1 + 1)$-dimensional (2D) free Abelian (and self-interacting non-Abelian) gauge theories, it turns out that the 4D free Abelian 2-form gauge theory is {\it not} an exact TFT. To corroborate this conclusion, some of the key issues are discussed. In particular, it is shown that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form Abelian gauge theory obey the recursion relations that are reminiscent of the exact TFTs but the Lagrangian density of this theory is not found to be able to be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as is the case with the {\it topological} 2D free Abelian (and self-interacting non-Abelian) gauge theories.

hep-th↗

arXiv2003

Cohomological aspects of gauge theories: superfield formalism

Some of the key cohomological features of the two $(1+1)$-dimensional (2D) free Abelian- and self-interacting non-Abelian gauge theories (having no interaction with matter fields) are briefly discussed first in the language of symmetry properties of the Lagrangian densities and the same issues are subsequently addressed in the framework of superfield formulation on the four ($2 + 2)$-dimensional supermanifold. Special emphasis is laid on the on-shell- and off-shell nilpotent (co-)BRST symmetries that emerge after the application of (dual) horizontality conditions on the supermanifold. The (anti-)chiral superfields play a very decisive role in the derivation of the on-shell nilpotent symmetries. The study of the present superfield formulation leads to the derivation of some new symmetries for the Lagrangian density and the symmetric energy-momentum tensor. The topological nature of the above theories is captured in the framework of superfield formulation and the geometrical interpretations are provided for some of the topologically interesting quantities.

hep-th↗

arXiv2002

Superfield approach to topological features of non-Abelian gauge theory

We discuss some of the key topological aspects of a two $(1+1)$-dimensional (2D) self-interacting non-Abelian gauge theory (having no interaction with matter fields) in the framework of {\it chiral} superfield formalism. We provide the geometrical interpretation for the Lagrangian density, symmetric energy momentum tensor, topological invariants, etc., by exploiting the {\it on-shell} nilpotent BRST- and co-BRST symmetries that emerge after the application of (dual) horizontality conditions. We show that the above physically interesting quantities geometrically correspond to the translation of some local (but composite) {\it chiral} superfields along one of the two independent Grassmannian directions of a four ($2+2)$-dimensional supermanifold. This translation is generated by the conserved and on-shell nilpotent (co-)BRST charges that are present in the theory.

hep-th↗