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.

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.

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.

Geometry and Quantum Dynamics

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
General Relativity ◽  
Quantum Dynamics ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Abelian Gauge ◽  
Yang Mills ◽  
History Of ◽  
Brst Invariance

A geometrical derivation of Abelian and non- Abelian gauge theories. The Faddeev–Popov quantisation. BRST invariance and ghost fields. General discussion of BRST symmetry. Application to Yang–Mills theories and general relativity. A brief history of gauge theories.

scholarly journals On gauge independence for gauge models with soft breaking of BRST symmetry

2014 ◽  
Vol 29 (30) ◽  
pp. 1450184 ◽  
Author(s):  
Alexander Reshetnyak
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Landau Gauge ◽  
Functional Integral ◽  
Yang Mills ◽  
Gauge Models ◽  
S Matrix ◽  
Gauge Fixing ◽  
Dependent Parameter ◽  
Quantum Treatment

A consistent quantum treatment of general gauge theories with an arbitrary gauge-fixing in the presence of soft breaking of the BRST symmetry in the field–antifield formalism is developed. It is based on a gauged (involving a field-dependent parameter) version of finite BRST transformations. The prescription allows one to restore the gauge-independence of the effective action at its extremals and therefore also that of the conventional S-matrix for a theory with BRST-breaking terms being additively introduced into a BRST-invariant action in order to achieve a consistency of the functional integral. We demonstrate the applicability of this prescription within the approach of functional renormalization group to the Yang–Mills and gravity theories. The Gribov–Zwanziger action and the refined Gribov–Zwanziger action for a many-parameter family of gauges, including the Coulomb, axial and covariant gauges, are derived perturbatively on the basis of finite gauged BRST transformations starting from Landau gauge. It is proved that gauge theories with soft breaking of BRST symmetry can be made consistent if the transformed BRST-breaking terms satisfy the same soft BRST symmetry breaking condition in the resulting gauge as the untransformed ones in the initial gauge, and also without this requirement.

scholarly journals LAGRANGIAN Sp(3) BRST SYMMETRY FOR IRREDUCIBLE GAUGE THEORIES

2001 ◽  
Vol 16 (17) ◽  
pp. 2975-3009 ◽  
Author(s):  
C. BIZDADEA ◽  
S. O. SALIU
Keyword(s):  
Perturbation Theory ◽  
Effective Action ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Gauge Fixing ◽  
Homological Perturbation Theory ◽  
Canonical Generator ◽  
Homological Perturbation

The Lagrangian Sp(3) BRST symmetry for irreducible gauge theories is constructed in the framework of homological perturbation theory. The canonical generator of this extended symmetry is shown to exist. A gauge-fixing procedure specific to the standard antibracket–antifield formalism, that leads to an effective action, which is invariant under all the three differentials of the Sp(3) algebra, is given.

scholarly journals BRST Charge and Poisson Algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
H. Caprasse
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Poisson Algebra ◽  
Global Symmetry ◽  
Nonlinear Case ◽  
Quadratic Algebras ◽  
Brst Charge ◽  
Gauge Fixing ◽  
International Audience ◽  
Formal Power

International audience An elementary introduction to the classical version of gauge theories is made. The shortcomings of the usual gauge fixing process are pointed out. They justify the need to replace it by a global symmetry: the BRST symmetry and its associated BRST charge. The main mathematical steps required to construct it are described. The algebra of constraints is, in general, a nonlinear Poisson algebra. In the nonlinear case the computation of the BRST charge by hand is hard. Itis explained how this computation can be made algorithmic. The main features of a recently created BRST computer algebra program are described. It can handle quadratic algebras very easily. Its capability to compute the BRST charge as a formal power series in the generic case of a cubic algebra is illustrated.

Quantum chromodynamics: A non-Abelian gauge theory

2019 ◽  
pp. 209-236
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Chromodynamics ◽  
Particle Physics ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Parallel Transport ◽  
Strong Interactions ◽  
Asymptotic Freedom ◽  
Abelian Gauge ◽  
Lattice Gauge ◽  
Colour Group

Chapter 13 is devoted to some aspects of quantum chromodynamics (QCD), the part of the Standard Model of particle physics responsible for strong interactions and based on an SU(3) gauge symmetry (the colour group) and gluon gauge fields. First, the geometry of non–Abelian gauge theories, based on parallel transport, is recalled. This leads naturally to the construction of lattice gauge theories with link variables and a plaquette action. The lattice model gives a hint of confinement. QCD is quantized in the temporal of Weyl gauge. Its renormalization involves the BRST symmetry. Its renormalization group properties with asymptotic freedom are emphasized. The infinite degeneracy of the semi–classical ground state can be associated to a winding number. Barrier penetration effects, related to the existence of instantons, lead to the existence of theta vacua and the problem of strong CP violation. Other issues considered are chiral symmetry and axial anomaly.

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.

Sign in / Sign up

Export Citation Format

Share Document