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.

Christ–Lee Model: (Anti-)chiral Supervariable Approach to BRST Formalism

We derive the off-shell nilpotent of order two and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations for the Christ–Lee (CL) model in one 0 + 1 -dimension (1D) of spacetime by exploiting the (anti-)chiral supervariable approach (ACSA) to BRST formalism where the quantum symmetry (i.e., (anti-)BRST along with (anti-)co-BRST) invariant quantities play a crucial role. We prove the nilpotency and absolute anticommutativity properties of the (anti-)BRST along with (anti-)co-BRST conserved charges within the scope of ACSA to BRST formalism where we take only one Grassmannian variable into account. We also show the (anti-)BRST and (anti-)co-BRST invariances of the Lagrangian within the scope of ACSA.

Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two (1[Formula: see text]+[Formula: see text]1)-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic 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 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.

BRST analysis and BFV quantization of the generalized quantum rigid rotor

We identify a strong similarity among several distinct originally second-class systems, including both mechanical and field theory models, which can be naturally described in a gauge-invariant way. The canonical structure of such related systems is encoded into a gauge-invariant generalization of the quantum rigid rotor. We perform the BRST symmetry analysis and the BFV functional quantization for the mentioned gauge-invariant version of the generalized quantum rigid rotor. We obtain different equivalent effective actions according to specific gauge-fixing choices, showing explicitly their BRST symmetries. We apply and exemplify the ideas discussed to two particular models, namely, motion along an elliptical path and the [Formula: see text] nonlinear sigma model, showing that our results reproduce and connect previously unrelated known gauge-invariant systems.

Langevin field equations: Properties and renormalization

Langevin equations for fields have been proposed to describe the dynamics of critical phenomena, or as an alternative method of quantization, which could be useful in cases where ordinary methods lead to difficulties, like in gauge theories. Some of their general properties will be described here. For a number of problems, in particular related to perturbation theory, it is more convenient to work with an action and a field integral than with the equation directly, because standard methods of quantum field theory (QFT) then become available. For this purpose, one can associate a field integral representation, involving a dynamic action to the Langevin equation. The dynamic action can be interpreted as generated by the Langevin equation, considered as a constraint equation. Quite generally, the integral representation of constraint equations, including stochastic equations, leads to an action that has a Slavnov–Taylor (ST) symmetry and, in a different form, has an anticommuting type Becchi–Rouet–Stora–Tyutin (BRST) symmetry, a symmetry that involves anticommuting parameters. This symmetry has no geometric origin, but is merely a consequence of associating a specific form of integral representations to the constraint equations. This symmetry is used in a number of different examples to prove the renormalizability of non-Abelian gauge theories, or to prove the geometric stability of models defined on homogeneous spaces under renormalization. In this chapter, it is used to prove Ward-Takahashi (WT) identities, and to determine how the Langevin equation renormalizes.