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.

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.

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.

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 THE MOST GENERAL AND RENORMALIZABLE MAXIMAL ABELIAN GAUGE

2003 ◽  
Vol 18 (31) ◽  
pp. 5733-5756 ◽  
Author(s):  
TORU SHINOHARA ◽  
TAKAHITO IMAI ◽  
KEI-ICHI KONDO
Keyword(s):  
Beta Function ◽  
Abelian Gauge ◽  
Yang Mills ◽  
Anomalous Dimensions ◽  
Diagonal Part ◽  
Gauge Term ◽  
Gauge Fixing ◽  
Popov Ghost ◽  
Independent Parameters ◽  
Energy Physics

We construct the most general gauge fixing and the associated Faddeev–Popov ghost term for the SU(2) Yang–Mills theory, which leaves the global U(1) gauge symmetry intact (i.e. the most general Maximal Abelian gauge). We show that the most general form involves eleven independent gauge parameters. Then we require various symmetries which help to reduce the number of independent parameters for obtaining the simpler form. In the simplest case, the off-diagonal part of the gauge fixing term obtained in this way is identical to the modified maximal Abelian gauge term with two gauge parameters which was proposed in the previous paper from the viewpoint of renormalizability. In this case, moreover, we calculate the beta function, anomalous dimensions of all fields and renormalization group functions of all gauge parameters in perturbation theory to one-loop order. We also discuss the implication of these results to obtain information on low-energy physics of QCD.

Non-Abelian gauge theories: Introduction

2021 ◽  
pp. 548-566
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Perturbation Theory ◽  
Quantum Electrodynamics ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Parallel Transport ◽  
Strong Interactions ◽  
Abelian Gauge ◽  
Temporal Gauge ◽  
Feynman Rules ◽  
Formal Properties

To be able to describe the other fundamental interactions, beyond quantum electrodynamics (QED), weak and strong interactions, it is necessary to generalize the concept of gauge symmetry to non-Abelian groups. Therefore, in this chapter, a quantum field theory (QFT)-invariant under local, that is, space-time-dependent, transformations of matrix representations of a general compact Lie groups are constructed. Inspired by the Abelian example, the geometric concept of parallel transport is introduced, a concept discussed more extensively later in the framework of Riemannian manifolds. All the required mathematical quantities for gauge theories then appear naturally. Gauge theories are quantized in the temporal gauge. The equivalence with covariant gauges is then established. Some formal properties of the quantized theory, like the Becchi–Rouet–Stora–Tyutin (BRST) symmetry, are derived. Feynman rules of perturbation theory are derived, the regularization of perturbation theory is discussed, a somewhat non-trivial problem. Some general properties of the non-Abelian Higgs mechanism are described.

scholarly journals CUBIC SUPERSYMMETRY AND ABELIAN GAUGE INVARIANCE

2005 ◽  
Vol 20 (25) ◽  
pp. 5779-5806 ◽  
Author(s):  
G. MOULTAKA ◽  
M. RAUSCH DE TRAUBENBERG ◽  
A. TANASĂ
Keyword(s):  
Gauge Symmetry ◽  
Gauge Invariance ◽  
Gauge Fields ◽  
Abelian Gauge ◽  
Interaction Terms ◽  
Gauge Fixing ◽  
Poincaré Symmetry

On the basis of recent results extending nontrivially the Poincaré symmetry, we investigate the properties of bosonic multiplets including 2-form gauge fields. Invariant-free Lagrangians are explicitly built which involve possibly 3- and 4-form fields. We also study in detail the interplay between this symmetry and a U (1) gauge symmetry, and in particular the implications of the automatic gauge-fixing of the latter corresponding to a residual gauge invariance, as well as the absence of self-interaction terms.

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.

Sign in / Sign up

Export Citation Format

Share Document