arXiv subjects

A. K. Rao

Publications and source records attributed to A. K. Rao.

15 recordsLinked to original sources

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..

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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.

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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.

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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.

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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.

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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.

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arXiv2022

Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review

In some of the physically interesting gauge systems, we show that the application of the Noether theorem does not lead to the deduction of the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST charges that obey precisely the off-shell nilpotency property despite the fact that these charges are $(i)$ derived by using the off-shell nilpotent (anti-)BRST symmetry transformations, $(ii)$ the generators of the above continuous symmetry transformations, and $(iii)$ conserved w.r.t. the time-evolution due to the Euler-Lagrange equations of motion derived from the Lagrangians/Lagrangian densities (that describe the dynamics of the suitably chosen physical systems). We propose a systematic method for the derivation of the off-shell nilpotent (anti-)BRST charges from the corresponding {non-nilpotent Noether conserved (anti-)BRST charges. To corroborate the sanctity and preciseness of our proposal, we take into account the examples of $(i)$ the one ($0 + 1$)-dimensional (1D) system of a massive spinning (i.e. SUSY) relativistic particle, $(ii)$ the D-dimensional non-Abelian 1-form gauge theory, and $(iii)$ the Abelian 2-form and the St${\ddot u}$ckelberg-modified version of the massive Abelian 3-form gauge theories in any arbitrary D-dimension of spacetime. Our present endeavor is a brief review where some decisive proposals have been made and a few novel results have been obtained as far as the nilpotency property is concerned.

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arXiv2022

1D Diffeomorphism Invariant Model of a Free Scalar Relativistic Particle: Supervariable and BRST Approaches

We apply the supervariable approach to derive the proper quantum Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries for the 1D diffeomorphism invariant model of a free scalar relativistic particle by exploiting the infinitesimal classical reparameterization (i.e. 1D diffeomorphism) symmetry of this theory. We derive the conserved and off-shell nilpotent (anti-)BRST charges and prove their absolute anticommutativity property by using the virtues of Curci-Ferrari (CF)-type restriction of our present theory. We establish the sanctity of the existence of CF-type restriction (i) by considering the (anti-)BRST symmetry transformations of the coupled (but equivalent) Lagrangians, and (ii) by proving the symmetry invariance of the Lagrangians within the framework of supervariable approach. We capture the nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of (anti-)chiral supervariable approach (ACSA) to BRST formalism. One of the novel observations of our present endeavor is the derivation of CF-type restriction by using the modified Bonora-Tonin (BT) supervariable approach (while deriving the (anti-)BRST symmetries for the target spacetime and/or momenta variables) and by symmetry considerations of the Lagrangians of the theory. The rest of the (anti-)BRST symmetries, for the other variables, are derived by using the newly proposed ACSA. We also demonstrate the existence of CF-type restriction in the proof of absolute anticommutativity of the (anti-)BRST charges.

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arXiv2022

Nilpotent Symmetries of a Modified Massive Abelian 3-Form Theory: Augmented Superfield Approach

We derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations for any arbitrary D-dimensional St$\ddot u$ckelberg-modified massive Abelian 3-form theory within the framework of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism where, in addition to the horizontality condition (HC), we exploit the theoretical strength of the gauge invariant restriction (GIR) to deduce the proper transformations for the gauge, associated (anti-)ghost fields, auxiliary fields, St$\ddot u$ckelberg compensating field, etc. In fact, it is an elegant and delicate combination of HC and GIR (within the ambit of AVSA) that is crucial for all our discussions and derivations. One of the highlights of our present endeavor is the deduction of a new set of (anti-)BRST invariant Curci-Ferrari (CF)-type restrictions which are not found in the massless version of our present theory where only the HC plays an important role in the derivation of all the (anti-)BRST transformations and a very specific set of CF-type restrictions. The alternative ways of the derivation of the full set of the latter, from various theoretical considerations, are also interesting results of our present investigation.

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arXiv2022

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari Type Restrictions

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two $(1 + 1)$-dimensional (2D) diffeomorphism symmetry transformations on the world-sheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari (CF) type restrictions (using MBTSA) which turn out to be the root-cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

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arXiv2021

Reparameterization Invariant Model of a Supersymmetric System: BRST and Supervariable Approaches

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one (0 + 1)-dimensional (1D) model of a free massive spinning relativistic particle (i.e. a supersymmetric system) by exploiting its classical infinitesimal and continuous reparameterization symmetry transformations. We use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the target space variables and the (anti-)BRST invariant Curci-Ferrari (CF)-type restriction for the 1D model of our supersymmetric (SUSY) system. The nilpotent (anti-)BRST symmetry transformations for other variables of our model are derived by using the (anti-)chiral supervariable approach (ACSA) to BRST formalism. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians, and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e. a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e. 1D diffeomorphism) invariant models of the (non-)relativistic particles.

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arXiv2021

Modified Proca Theory in Arbitrary and Two Dimensions

We demonstrate that the standard Stueckelberg-modified Proca theory (i.e. a massive Abelian 1-form theory) respects the classical gauge and corresponding quantum (anti-)BRST symmetry transformations in any arbitrary dimension of spacetime within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We further show that the Stueckelberg formalism gets modified in the two (1+1)-dimensions of spacetime due to a couple of discrete duality symmetries in the theory which turn out to be responsible for the existence of the nilpotent (anti-)co-BRST symmetry transformations corresponding to the nilpotent (anti-)BRST symmetry transformations of our theory. These nilpotent symmetries exist together in the modified version of the two (1+1)-dimensional (2D) Proca theory. We provide the mathematical basis for the modification of the Stueckelberg-technique, the existence of the discrete duality as well as the continuous (anti-)co-BRST symmetry transformations in the 2D modified version of Proca theory.

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arXiv2021

Supervariable and BRST Approaches to a Reparameterization Invariant Non-Relativistic System

We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e. off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a non-relativistic (NR) free particle whose space $(x)$ and time $(t)$ variables are function of an evolution parameter $(τ)$. The infinitesimal reparameterization (i.e. 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter $(τ)$. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti-)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by $τ$) is generalized onto a $(1, 2)$-dimensional supermanifold which is characterized by the superspace coordinates $Z^M = (τ, θ, \barθ)$ where a pair of Grassmannian variables satisfy the fermionic relationships: $θ^2 = {\barθ}^2 = 0, \, θ\,\barθ+ \barθ\,θ= 0$ and $τ$ is the bosonic evolution parameter. In the context of ACSA, we take into account only the (1, 1)-dimensional (anti-)chiral super sub-manifolds of the general (1, 2)-dimensional supermanifold. The derivation of the universal Curci-Ferrari (CF)-type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly same as that of the reparameterization invariant SUSY (i.e. spinning) and non-SUSY (i.e. scalar) relativistic particles. This is a novel observation, too.

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arXiv2021

Interacting 2D Field-Theoretic Model for Hodge Theory

We take up the St${\ddot u}$ckelberg-modified version of the two (1+1)-dimensional (2D) Proca theory, in interaction with the Dirac fields, to study its various continuous and discrete symmetry transformations and show that this specific interacting 2D field-theoretic model provides a tractable example for the Hodge theory because its symmetries (and corresponding conserved charges) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. The physical state of this theory is chosen to be the harmonic state (of the Hodge decomposed state) in the quantum Hilbert space which is annihilated by the conserved and nilpotent (anti-)BRST as well as (anti-)co-BRST charges. A physical consequence of this study is an observation that the 2D anomaly, at the quantum level, does not lead to any problem as far as the consistency and unitarity of our present 2D theory is concerned. In other words, our present 2D field-theoretic model is amenable to particle interpretation despite the presence of the local chiral symmetry (which is associated with the nilpotent (anti-)co-BRST symmetry transformations) besides the presence of the nilpotent (anti-)BRST symmetries (which are connected with the local gauge symmetry). The physicality condition with the (anti-)co-BRST charges implies that the 2D anomaly term is trivial in our present theory. Hence, our 2D theory is consistent, unitary and amenable to particle interpretation.

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arXiv2020

Massive Spinning Relativistic Particle: Revisited Under BRST and Supervariable Approaches

We discuss the continuous and infinitesimal gauge, supergauge, reparameterization, nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries and derive corresponding nilpotent charges for the one (0+1)-dimensional (1D) massive model of a spinning relativistic particle. We exploit the theoretical potential and power of the BRST and supervariable approaches to derive the (anti-)BRST symmetries and coupled (but equivalent) Lagrangians for this system. In particular, we capture the off-shell nilpotency and absolute anticommutatvity of the conserved (anti-)BRST charges within the framework of the newly proposed (anti-)chiral supervariable approach (ACSA) to BRST formalism where only the (anti-)chiral supervariables (and their suitable super expansions are taken into account along the Grassmannian direction(s)). One of the novel observations of our present investigation is the derivation of the Curci-Ferrari (CF)-type restriction by the requirement of the absolute anticommutatvity of the (anti-)BRST charges in the ordinary space. We obtain the same restriction within the framework of ACSA to BRST formalism by (i) the symmetry invariance of the coupled Lagrangians, and (ii) the proof of the absolute anticommutatvity of the conserved and nilpotent (anti-)BRST charges. The observation of the anticommutativity property of the (anti-)BRST charges is a novel result in view of the fact that we have taken into account only the (anti-)chiral super expansions.

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