scholarly journals Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
A. K. Rao ◽  
A. Tripathi ◽  
R. P. Malik
Keyword(s):  
Free Particle ◽  
Brst Symmetry ◽  
Symmetry Transformation ◽  
Present System ◽  
Evolution Parameter ◽  
Gauge Fixing ◽  
Symmetry Transformations ◽  
Diffeomorphism Symmetry ◽  
Popov Ghost ◽  
Relativistic Particles

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 nonrelativistic (NR) free particle whose space x and time t variables are a 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 , θ ¯ where a pair of the Grassmannian variables satisfy the fermionic relationships: θ 2 = θ ¯ 2 = 0 , θ   θ ¯ + θ ¯   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 submanifolds 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 the 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.

scholarly journals Nilpotent charges of a toy model of Hodge theory and an 𝒩 = 2 SUSY quantum mechanical model: (Anti-)chiral supervariable approach

2019 ◽  
Vol 34 (30) ◽  
pp. 1950183
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
General Formula ◽  
Mechanical Model ◽  
Free Particle ◽  
Brst Symmetry ◽  
Hodge Theory ◽  
Quantum Mechanical ◽  
Rigid Rotor ◽  
Quantum Mechanical Model ◽  
Evolution Parameter ◽  
Symmetry Transformations

We derive the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the system of a toy model of Hodge theory (i.e. a rigid rotor) by exploiting the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral supervariables that are defined on the appropriately chosen [Formula: see text]-dimensional super-submanifolds of the general [Formula: see text]-dimensional supermanifold on which our system of a one [Formula: see text]-dimensional (1D) toy model of Hodge theory is considered within the framework of the augmented version of the (anti-)chiral supervariable approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism. The general [Formula: see text]-dimensional supermanifold is parametrized by the superspace coordinates [Formula: see text], where [Formula: see text] is the bosonic evolution parameter and [Formula: see text] are the Grassmannian variables which obey the standard fermionic relationships: [Formula: see text], [Formula: see text]. We provide the geometrical interpretations for the symmetry invariance and nilpotency property. Furthermore, in our present endeavor, we establish the property of absolute anticommutativity of the conserved fermionic charges which is a completely novel and surprising observation in our present endeavor where we have considered only the (anti-)chiral supervariables. To corroborate the novelty of the above observation, we apply this ACSA to an [Formula: see text] SUSY quantum mechanical (QM) system of a free particle and show that the [Formula: see text] SUSY conserved and nilpotent charges do not absolutely anticommute.

From BRST symmetry to the Zinn-Justin equation

2019 ◽  
pp. 237-252
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Quadratic Equation ◽  
Local Action ◽  
Local Form ◽  
Abelian Gauge ◽  
Brst Cohomology ◽  
Gauge Fixing ◽  
Non Local ◽  
Popov Ghost

Chapter 14 contains a general discussion of the quantization and renormalization of non–Abelian gauge theories. The quantization necessitates gauge fixing and introduces the Faddeev–Popov determinant. Slavnov–Taylor identities for vertex (one–particle–irreducible (1PI)) functions, the basis of a first proof of renormalizability, follow. The Faddeev–Popov determinant leads to a non–local action. A local form is generated by introducing Faddeev–Popov ghost fields. The new local action has an important new symmetry, the BRST symmetry. However, the explicit realization of the symmetry is not stable under renormalization. By contrast, a quadratic equation that is satisfied by the action and generating functional of 1PI functions, the Zinn–Justin equation, is stable and at the basis of a general proof of the renormalizability of non–Abelian gauge theories. The proof involves some simple elements of BRST cohomology. The renormalized form of BRST symmetry then makes it possible to prove gauge independence and unitarity.

Gauge invariance and gauge fixing

2019 ◽  
pp. 177-194
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Gauge Invariance ◽  
Brst Symmetry ◽  
Parallel Transport ◽  
Abelian Gauge ◽  
The Standard Model ◽  
Gauge Fixing ◽  
Popov Ghost ◽  
Important Notion ◽  
Gribov Copies

Chapter 11 is the first of four chapters that discuss various issues connected with the Standard Model of fundamental interactions at the microscopic scale. It discusses the important notion of gauge invariance, first Abelian and then non–Abelian, the basic geometric structure that generates interactions. It relates it to the concept of parallel transport. Due to gauge invariance, not all components of the gauge field are dynamical and gauge fixing is required (with the problem of Gribov copies in non–Abelian theories). The quantization of non–Abelian gauge theories is briefly discussed, with the introduction of Faddeev–Popov ghost fields and the appearance of BRST symmetry.

Becchi–Rouet–Stora–Tyutin (BRST) symmetry. Gauge theories: Zinn-Justin equation (ZJ) and renormalization

2021 ◽  
pp. 623-655
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Differential Operator ◽  
Simple Form ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Quantization Method ◽  
Abelian Gauge ◽  
Gauge Action ◽  
Temporal Gauge ◽  
Gauge Fixing ◽  
Popov Ghost

The first part of the chapter describes Faddeev–Popov's quantization method, nd the resulting Slavnov–Taylor (ST) identities, in a simple context. This construction automatically implies, after introduction of Faddeev–Popov ‘ghost’ fermions, a Becchi–Rouet–Stora–Tyutin (BRST) symmetry, whose properties are derived. The differential operator, of fermionic type, representing the BRST symmetry, with a proper choice of variables, has the form of a cohomology operator, and a simple form in terms of Grassmann coordinates. The second part of the chapter is devoted to the quantization and renormalization of non-Abelian gauge theories. Quantization of gauge theories require a gauge-fixing procedure. Starting from the non-covariant temporal gauge, and using a simple identity, one shows the equivalence with a quantization in a general class of gauges, including relativistic covariant gauges. Adapting the formalism developed in the first part, ST identities, and the corresponding BRST symmetry are derived. However, the explicit form of the BRST symmetry is not stable under renormalization. The BRST symmetry implies a more general, quadratic master equation, also called Zinn-Justin (ZJ) equation, satisfied by the quantized action, equation in which gauge and BRST symmetries are no longer explicit. By contrast, in the case of renormalizable gauges, the ZJ equation is stable under renormalization, and its solution yields the general form of the renormalized gauge action.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
A. Tripathi ◽  
B. Chauhan ◽  
A. K. Rao ◽  
R. P. Malik
Keyword(s):  
Brst Quantization ◽  
Relativistic Particle ◽  
Brst Symmetry ◽  
Target Space ◽  
Spinning Particle ◽  
1D Model ◽  
Symmetry Transformations ◽  
The Absolute ◽  
The One ◽  
Relativistic Particles

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.

scholarly journals NILPOTENT SYMMETRIES FOR A FREE RELATIVISTIC PARTICLE IN AUGMENTED SUPERFIELD FORMALISM

2005 ◽  
Vol 20 (23) ◽  
pp. 1767-1779 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Relativistic Particle ◽  
World Line ◽  
Free Particle ◽  
Brst Symmetry ◽  
Dimensional System ◽  
Gravitational Theories ◽  
Target Manifold ◽  
Free Scalar ◽  
Symmetry Transformations ◽  
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 [Formula: see text]) 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.

scholarly journals DUAL BRST SYMMETRY FOR QED

2001 ◽  
Vol 16 (08) ◽  
pp. 477-488 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Brst Symmetry ◽  
Discrete Symmetry ◽  
Two Dimensions ◽  
Abelian Gauge ◽  
Dirac Fields ◽  
Gauge Fixing ◽  
De Rham ◽  
Symmetry Transformations ◽  
Dimensions Of Space ◽  
Field Theoretical Model

We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space–time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry.

scholarly journals Novel symmetries in vector Schwinger model

2014 ◽  
Vol 29 (15) ◽  
pp. 1450076 ◽  
Author(s):  
Saurabh Gupta
Keyword(s):  
Differential Geometry ◽  
Brst Symmetry ◽  
Laplacian Operator ◽  
Exterior Derivative ◽  
Schwinger Model ◽  
Gauge Fixing ◽  
De Rham ◽  
Symmetry Transformations

We derive nilpotent and absolutely anticommuting (anti-)co-BRST symmetry transformations for the bosonized version of (1+1)-dimensional (2D) vector Schwinger model. These symmetry transformations turn out to be the analog of co-exterior derivative of differential geometry as the total gauge-fixing term remains invariant under it. The exterior derivative is realized in terms of the (anti-)BRST symmetry transformations of the theory whereas the bosonic symmetries find their analog in the Laplacian operator. The algebra obeyed by these symmetry transformations turns out to be exactly same as the algebra obeyed by the de Rham cohomological operators of differential geometry.

scholarly journals GAUGE FIXING AND OBSERVABLES IN GENERAL RELATIVITY

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2475-2482 ◽  
Author(s):  
DONALD SALISBURY
Keyword(s):  
General Relativity ◽  
Transformation Group ◽  
Proper Time ◽  
Free Particle ◽  
Symmetry Transformation ◽  
Time Dependent ◽  
Canonical Variables ◽  
Time Translation ◽  
Field Dependent ◽  
Gauge Fixing

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.

scholarly journals Faddeev-Popov Ghost and BRST Symmetry in Yang-Mills Theory

2020 ◽  
Vol 31 (1) ◽  
pp. 30-34
Author(s):  
Edyharto Yanuwar ◽  
Jusak Sali Kosasih
Keyword(s):  
Gauge Field ◽  
Gauge Transformation ◽  
Brst Symmetry ◽  
Brst Transformation ◽  
Least Action Principle ◽  
Yang Mills ◽  
Least Action ◽  
Gauge Fixing ◽  
Popov Ghost ◽  
Ghost Field

Ghost fields arise from the quantization of the gauge field with constraints (gauge fixing) through the path integral method. By substituting a form of identity, an effective propagator will be obtained from the gauge field with constraints and this is called the Faddeev-Popov method. The Grassmann odd properties of the ghost field cause the gauge transformation parameter to be Grassmann odd, so a BRST transformation is defined. Ghost field emergence with Grassmann odd properties can also be obtained through the least action principle with gauge transformation, and thus the relations between the BRST transformation parameters and the ghost field is obtained.

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