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.

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.

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.

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 A resource efficient approach for quantum and classical simulations of gauge theories in particle physics

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 393
Author(s):  
Jan F. Haase ◽  
Luca Dellantonio ◽  
Alessio Celi ◽  
Danny Paulson ◽  
Angus Kan ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Quantum Electrodynamics ◽  
Particle Physics ◽  
Gauge Theories ◽  
Hamiltonian Formulation ◽  
Mcmc Methods ◽  
Continuous Limit ◽  
Abelian Gauge ◽  
Gauge Groups

Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2+1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.

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.

scholarly journals EXACT RENORMALIZATION GROUP IN ALGEBRAIC NONCOVARIANT GAUGES

2001 ◽  
Vol 16 (11) ◽  
pp. 2125-2130
Author(s):  
M. SIMIONATO
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Light Cone ◽  
Wilson Loop ◽  
Gauge Theories ◽  
Mass Term ◽  
Abelian Gauge ◽  
Light Cone Gauge ◽  
Infrared Limit ◽  
Exact Renormalization Group

I study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff Λ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless the BRS-invariance in broken and the theory is gauge-dependent and unphysical at Λ≠ 0. Then I discuss the infrared limit Λ→0. I show that the singularities of the axial gauge choice are avoided in planar gauge and in light-cone gauge. Finally the rectangular Wilson loop of size 2L×2T is evaluated at lowest order in perturbation theory and a noncommutativity between the limits Λ→0 and T→∞ is pointed out.

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.

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