arXiv subjects

R. P. Malik

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

At least 19 recordsLinked to original sources

arXiv2026

Canonical Brackets, (Anti-)BRST Invariant Charges and Physicality Criteria: Non-Abelian 1-Form Gauge Theory

In the case of a D-dimensional non-Abelian 1-form gauge theory (without any interaction with the matter fields), we show that the application of the Noether theorem does not lead to the derivations of the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST charges that obey (i) the (anti-)BRST invariance, and (ii) the nilpotency property (unless we exploit the theoretical strength of $(a)$ the partial integration along with the Gauss divergence theorem, and $(b)$ use the appropriate equations of motion at suitable places). This happens because of the presence of a non-trivial Curci-Ferrari (CF) condition on our BRST-quantized D-dimensional non-Abelian 1-form gauge theory (whose limiting case is the BRST-quantized D-dimensional Abelian 1-form gauge theory where the CF-type restriction is trivial and the corresponding Noether (anti-)BRST charges turn out to be nilpotent as well as (anti-)BRST invariant quantities together). We exploit the theoretical strength of the basic canonical (anti)commutators to prove (i) the Noether (anti-)BRST charges as the true generators for the off-shell nilpotent (anti-)BRST symmetry transformations, and (ii) the (anti-)BRST invariance of the consistently modified versions of the Noether (anti-)BRST charges. We demonstrate that the (anti-)BRST invariant versions of the conserved (anti-)BRST charges are useful in the discussion on the physicality criteria because of their consistency with the Dirac quantization conditions for the gauge theories.

hep-th↗

arXiv2026

Abelian 2-Form Gauge Theory: Basic Canonical Brackets and Nilpotency Property of the Noether (Anti-)BRST Charges

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we invoke the beauty of the basic canonical (anti)commutators to prove the nilpotency property of the Noether (anti-)BRST charges for the D-dimensional BRST-quantized version of the free Abelian 2-form gauge theory which is endowed with a non-trivial Curci-Ferrari (CF) type restriction. In this proof, we use only the theoretical strength of the Gauss divergence theorem. We demonstrate that, under the off-shell nilpotent (anti-)BRST symmetry transformations, the Noether conserved (anti-)BRST charges are not invariant and they are also found to be not off-shell nilpotent (if we exploit the standard relationship between the continuous symmetry transformations and their generators as the Noether conserved charges). However, these charges become (anti-)BRST invariant and nilpotent if we use (i) the appropriate equations of motion at suitable places, and (ii) the Gauss divergence theorem. We derive the consistently modified versions of the Noether (anti-)BRST charges which are invariant under the off-shell nilpotent (anti-)BRST transformations. We prove the (anti-)BRST invariance of these modified versions of charges by using the basic canonical (anti)commutators, too. We discuss the physicality criteria w.r.t. (i) the conserved Noether (anti-)BRST charges, and (ii) the modified (anti-)BRST invariant versions of the Noether (anti-)BRST charges. We prove the superiority of the latter over the former (in view of the consistency with the Dirac quantization conditions for the gauge theories).

hep-th↗

arXiv2026

A Unique Bosonic Symmetry in a 4D Field-Theoretic System

For the combined field-theoretic system of the four (3 + 1)-dimensional (4D) Abelian 3-form and 1-form gauge theories, we show the existence of a unique bosonic symmetry transformation that is constructed from the four infinitesimal, continuous and off-shell nilpotent symmetry transformations which exist for the Becchi-Rouet-Stora-Tyutin (BRST) quantized versions of the coupled (but equivalent) Lagrangian densities that describe our present 4D field-theoretic system. The above off-shell nilpotent symmetry transformations are nothing but the BRST, co-BRST, anti-BRST and anti-co-BRST, under which, the Lagrangian densities transform to the total spacetime derivatives. The proof of the uniqueness of the above bosonic symmetry transformation operator crucially depends on the validity of all the four Curci-Ferrari (CF) type restrictions that exist on our theory. We highlight the importance of these CF-type restrictions, at various levels of our theoretical discussions, in the context of the unique bosonic symmetry transformation operator. We compare this observation against the backdrop of the three CF-type restrictions that appear in the requirements of the absolute anticommutativity between the specific set of a couple of nilpotent symmetry transformation operators.

hep-th↗

arXiv2026

Discrete and Continuous Symmetry Transformation Operators and Their Algebraic Structures: A $3D$ Field-Theoretic System

We discuss the discrete as well as the continuous symmetry transformations for a three $(2+1)$-dimensional $(3D)$ combined system of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism and establish their relevance in the context of the algebraic structures that are obeyed by the de Rham cohomological operators of differential geometry. In fact, our present field-theoretic system respects six continuous symmetry transformations and a couple of very useful discrete duality symmetry transformations. Out of the above six continuous symmetry transformations four are off-shell nilpotent (i.e. fermionic) in nature and two are bosonic. The algebraic structures, obeyed by the symmetry operators, are reminiscent of the algebra satisfied by the de Rham cohomological operators. Hence, our present $3D$ field-theoretic system provides a perfect example for Hodge theory where there is convergence of ideas from the physical aspects of the BRST formalism and mathematical ingredients that are connected with the cohomological operators of differential geometry at the algebraic level. One of the highlights of our present investigation is the appearance of a pseudo-scalar field in our theory (on the symmetry ground alone) which carries the negative kinetic term. Thus, it is one of the possible candidates for the ``phantom" fields of the cyclic, bouncing and self-accelerated cosmological models of the Universe.

hep-th↗

arXiv2025

Nilpotency Property, Physicality Criteria, Constraints and Standard BRST Algebra: A 4D Field-Theoretic System

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the off-shell nilpotent (anti-)BRST and the bosonic ghost-scale symmetries of a set of coupled (but equivalent) Lagrangian densities for the four (3 + 1)-dimensional (4D) combined field-theoretic system of the free Abelian 3-form and 1-form gauge theories. We demonstrate that the Noether (anti-)BRST charges are non-nilpotent due to the presence of a set of non-trivial Curci-Ferrari (CF) type restrictions on our theory. These CF-type restrictions are derived and discussed from different theoretical angles in our present endeavor. In addition to it, we obtain the nilpotent versions of the above (anti-)BRST charges from their counterparts non-nilpotent versions and discuss the physicality criteria w.r.t. the nilpotent charges to show that the physical states (existing in the total quantum Hilbert space of states) are those that are annihilated by the operator forms of the first-class constraints of our classical 4D combined field-theoretic system. The standard BRST algebra among the nilpotent (anti-)BRST charges and the bosonic ghost charge is derived, too.

hep-th↗

arXiv2025

A $3D$ Field-Theoretic Model: Discrete Duality Symmetry

We demonstrate the discrete duality symmetry between the Abelian 1-form and 2-form basic gauge fields in the context of a three $(2 + 1)$-dimensional ($3D$) combined system of the field-theoretic model of the free Abelian 1-from and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. The classical gauge-fixed Lagrangian density of this theory is generalized to its quantum counterpart as the BRST and co-BRST invariant Lagrangian density. We show clearly the existence of the off-shell nilpotent (co-)BRST symmetry transformations and establish their intimate connection through a set of underlying discrete duality symmetry transformations in our $3D$ BRST-quantized theory. We provide the mathematical basis for the existence of the discrete duality symmetry transformations in our theory through the Hodge duality operator (that is defined on the $3D$ flat Minkowskian spacetime manifold). We briefly mention a bosonic symmetry transformation which is constructed from the anticommutator of the above off-shell nilpotent (co-)BRST symmetry transformations. We lay emphasis on the algebraic structures of the existing continuous and discrete duality symmetry transformations for our $3D$ BRST-quantized theory (where they are treated as operators). We also comment on the appearance of a pseudo-scalar field (with negative kinetic term). This field happens to be one of the possible candidates for the phantom field of the cosmological models.

hep-th↗

arXiv2025

Continuous and Discrete Symmetries in a 4D Field-Theoretic Model: Symmetry Operators and Their Algebraic Structures

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we show the existence of (i) a couple of off-shell nilpotent (i.e. fermionic) BRST and co-BRST symmetry transformations, and (ii) a full set of non-nilpotent (i.e. bosonic) symmetry transformations for an appropriate Lagrangian density that describes the combined system of the free Abelian 3-form and 1-form gauge theories in the physical four (3 + 1)-dimensions of the flat Minkowskian spacetime. This combined BRST-quantized field-theoretic system is essential for the existence of the off-shell nilpotent co-BRST and non-nilpotent bosonic symmetry transformations in the theory. We concentrate on the full algebraic structures of the above continuous symmetry transformation operators along with a couple of very useful discrete duality symmetry transformation operators existing in our four (3 + 1)-dimensional (4D) field-theoretic model. We establish the relevance of the algebraic structures, respected by the above discrete and continuous symmetry operators, to the algebraic structures that are obeyed by the de Rham cohomological operators of differential geometry. One of the highlights of our present endeavor is the observation that there are no ``exotic'' fields with the negative kinetic terms in our present 4D field-theoretic example for Hodge theory.

hep-th↗

arXiv2025

Modified 3D Massive Abelian 2-From Theory with a Single Pseudo-Scalar Field as a Phantom Field: BRST Approach

We obtain the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations (corresponding to the infinitesimal classical gauge symmetry transformations) for the modified massive three $(2+1)$-dimensional (3D) Abelian 2-form gauge theory with a single pseudo-scalar field. The latter field (having the negative kinetic term and a well-defined mass) has already been shown (i) to exist in the modified version of the standard 3D St${\ddot u}$ckelberg formalism (on the solid mathematical grounds), (ii) to be a possible candidate for the ``phantom" field of some of the cosmological models of the Universe, and (iii) to be a possible candidate for dark matter. A couple of highlights of our present endeavor are (i) the observation that, even though the pseudo-scalar field does not transform under the gauge and (anti-)BRST symmetry transformations, it appears in the first-class constraints which annihilate the physical states at the quantum level, and (ii) the Noether conserved (anti-)BRST charges are found to be non-nilpotent. In our present investigation, we derive (i) the coupled (but equivalent) BRST and anti-BRST invariant Lagrangian densities, (ii) the conserved and off-shell nilpotent versions of the (anti-)BRST charges and the conserved ghost charge, (iii) the (anti-)BRST invariant Curci-Ferrari (CF) type restrictions, and (iv) the standard BRST algebra amongst the conserved and nilpotent (anti-)BRST charges and conserved ghost charge of our theory.

hep-th↗

arXiv2025

Pseudo-Scalar Field as a Possible Candidate for Phantom Field

We demonstrate the existence of a single pseudo-scalar (PS) field in the mathematically backed and parity preserving modifications of the standard St${\ddot u}$ckelberg formalism (SSF) in the context of the Lagrangian formulation of the (i) two (1 + 1)-dimensional (2D) massive Abelian 1-form gauge theory, (ii) three (2 + 1)-dimensional (3D) massive Abelian 2-form gauge theory, and (iii) four (3 + 1)-dimensional (4D) massive Abelian 3-form gauge theory. This PS field always turns-up with the (i) negative kinetic term, and (ii) higher order derivative terms. The latter terms are rendered into the well-defined terms due to the imposition of the on-shell condition for the PS field which is derived from the properly gauge-fixed Lagrangian densities for the above theories. These Lagrangian densities incorporate into themselves the (i) PS field with the negative kinetic term, and (ii) pure scalar field with the positive kinetic term. However, both these fields satisfy the Klein-Gordon (KG) equation of motion. The PS field (having the negative kinetic term and a well-defined mass) is one of the possible candidates for the ``phantom'' field which plays a crucial role in the cyclic, bouncing and self-accelerated cosmological models of the Universe. The ``exotic'' PS field also satisfies one of the criteria for being a possible candidate for dark matter

hep-th↗

arXiv2024

Constraints, Conserved Charges and Extended BRST Algebra for a 3D Field-Theoretic Example for Hodge Theory

We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory $(i)$ at the classical level within the ambit of Lagrangian formulation, and $(ii)$ at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles.

hep-th↗

arXiv2024

BRST and Related Superfield Approach to a Few Interesting Models of Point Particles and a Model of Bosonic String: A Brief Review

In this brief review-cum-research article, we discuss a few key aspects of the off-shell nilpotent (anti-)BRST symmetry transformations, Curci-Ferrari (CF) type restriction(s), coupled Lagrangians/Lagrangian densities, etc., for the 1D diffeomorphism (i.e. reparameterization) invariant models of $(i)$ a non-relativistic and non-SUSY free particle, $(ii)$ a scalar (i.e. non-SUSY) relativistic free particle, $(iii)$ a spinning (i.e. SUSY) relativistic free particle, and $(iv)$ a 2D diffeomorphism invariant model of a specific bosonic string theory within the framework of BRST and related supervariable/superfield approach. We take up a new 1D diffeomorphism invariant model of an interacting scalar relativistic particle with the electromagnetic field. The latter is treated as a constant background variable and is, therefore, independent of the evolution parameter. We show that the universal nature of the CF-type restriction is maintained in the case of this new model, too. We exploit the modified Bonora-Tonin supervariable/superfield approach (MBTSA) to BRST formalism in the context of the 1D and 2D diffeomorphism invariant theories to prove the universal nature of the CF-type restriction(s). We further demonstrate that the 1D diffeomorphism invariant models of all kinds of particles are described by the singular Lagrangians. This singular nature has, to the best of our knowledge, not been explicitly shown elsewhere. Thus, these 1D models also respect the gauge symmetry transformations which are generated by the first-class constraints that exist on them. The equivalence of the 1D diffeomorphism and gauge symmetries is also established under specific conditions..

hep-th↗

arXiv2024

St$\ddot u$ckelberg-Modified Massive Abelian 3-Form Theory: Constraint Analysis, Conserved Charges and BRST Algebra

For the St$\ddot u$ckelberg-modified massive Abelian 3-form theory in any arbitrary D-dimension of spacetime, we show that its classical gauge symmetry transformations are generated by the first-class constraints. We establish that the Noether conserved charge (corresponding to the local gauge symmetry transformations) is same as the standard form of the generator for the underlying local gauge symmetry transformations (expressed in terms of the first-class constraints). We promote these classical local, continuous and infinitesimal gauge symmetry transformations to their quantum counterparts Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations which are respected by the coupled (but equivalent) Lagrangian densities. We derive the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether's theorem. However, these charges turn our to be non-nilpotent. Some of the highlights of our present investigation are (i) the derivation of the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the operator forms of the first-class constraints at the quantum level through the physicality criteria w.r.t. the nilpotent versions of the (anti-)BRST charges, and (iii) the deduction of the CF-type restrictions from the straightforward equality of the coupled (anti-)BRST invariant Lagrangian densities as well as from the requirement of the absolute anticommutativity of the off-shell nilpotent versions of the (anti-)BRST charges.

hep-th↗

arXiv2024

A 3D Field-Theoretic Example for Hodge Theory

We focus on the continuous symmetry transformations for the three ($2 + 1$)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present theory provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e. $D = 3$) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting ($0 + 1$)-dimensional toy models as well as a set of well-known ${\mathcal N} = 2$ SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass and gauge-invariance to co-exist together.

hep-th↗

arXiv2024

A Quantum Mechanical Example for Hodge Theory

On the basis of (i) the discrete and continuous symmetries (and corresponding conserved charges), (ii) the ensuing algebraic structures of the symmetry operators and conserved charges, and (iii) a few basic concepts behind the subject of differential geometry, we show that the celebrated Friedberg-Lee-Pang-Ren (FLPR) quantum mechanical model (describing the motion of a single non-relativistic particle of unit mass under the influence of the general spatial 2D rotationally invariant potential) provides a tractable physical example for the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the symmetry operators and conserved charges lead to the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. We concisely mention the Hodge decomposition theorem in the quantum Hilbert space of states and choose the harmonic states as the real physical states of our theory. We discuss the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST and (anti-)co-BRST charges and the physical consequences that ensue from them.

hep-th↗

arXiv2024

Constraints and Conserved Charges for Modified Massive and Massless Abelian 1-Form and 2-Form Theories: A Brief Review

We demonstrate that the generators for the local, continuous and infinitesimal classical gauge symmetry transformations in the cases of (i) the St$\ddot u$ckelberg-modified massive Abelian 1-form and 2-form theories, and (ii) the massless Abelian 1-form and 2-form free theories owe their origin to the first-class constraints of the these theories. We establish a connection between the standard forms of the generators and the Noether conserved charges for the modified massive and massless versions of the above theories. We discuss the appearance of these constraints, within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, through the physicality criteria w.r.t. the conserved and nilpotent (anti-)BRST charges. One of the highlights of our present investigation is the observation that, in the context of the modified massive and massless Abelian 2-form theories, the modified forms of the standard Noether (anti-)BRST charges are required which are found to be off-shell nilpotent and they lead to the appearance of the operator forms of the first-class constraints through the physicality criteria at the quantum level. We also comment on (i) the existence of the Curci-Ferrari (CF)-type restrictions on the Abelian 2-form theories (with and without mass), (ii) the modifications in the St$\ddot u$ckelberg-technique for the massive 2D Abelian 1-form and 4D Abelian 2-form theories and their consequences, and (iii) the off-shell nilpotent version of the conserved co-BRST charge and its role in the physicality criteria for the St$\ddot u$ckelberg-modified 4D massive Abelian 2-form theory.

hep-th↗

arXiv2023

FLPR Model: Nilpotent (Anti-)co-BRST Symmetries

We demonstrate the existence of a set of novel off-shell nilpotent and absolutely anticommuting continuous symmetry transformations, within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, which are over and above the usual off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations that are respected by the first-order Lagrangian for the Friedberg-Lee-Pang-Ren (FLPR) model that describes the motion of a non-relativistic particle of unit mass moving under the influence of a general rotationally invariant spatial two-dimensional potential. We christen these novel set of fermionic symmetry transformations as the (anti-)co-BRST symmetry transformations because the gauge-fixing term remains invariant under them. We derive the conserved and off-shell nilpotent (anti-)BRST and (anti-)co-BRST charges and comment on the physicality criteria w.r.t. them where we establish the presence of the operator forms of the first-class constraints (of the original classical gauge theory) at the quantum level.

hep-th↗

arXiv2023

Constraints, Symmetry Transformations and Conserved Charges for Massless Abelian 3-Form Theory

We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too.

hep-th↗

arXiv2023

Modified St$\ddot u$ckelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated St$\ddot u$ckelberg formalism gets modified in the case of a massive four (3+1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where there is existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual St$\ddot u$ckelberg-modified massive Abelian 2-form gauge theory. The modification in the St$\ddot u$ckelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the St$\ddot u$ckelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick to get rid of the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent and renormalizable.

hep-th↗