scholarly journals Generalization of Faddeev–Popov rules in Yang–Mills theories: N = 3,4 BRST symmetries

2018 ◽  
Vol 33 (03) ◽  
pp. 1850006 ◽  
Author(s):  
Alexander Reshetnyak
Keyword(s):  
Irreducible Representation ◽  
Classical Model ◽  
Brst Symmetry ◽  
Classical Action ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Change Of Variables ◽  
Gauge Fixing ◽  
Quantum Action

The Faddeev–Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, [Formula: see text], with respect to [Formula: see text]-parametric Abelian SUSY transformations with odd-valued parameters [Formula: see text], [Formula: see text] and generators [Formula: see text]: [Formula: see text], for [Formula: see text], implying the substitution of an [Formula: see text]-plet of ghost fields, [Formula: see text], instead of the parameter, [Formula: see text], of infinitesimal gauge transformations: [Formula: see text]. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the [Formula: see text] and [Formula: see text] symmetric cases. The superspace of [Formula: see text] SUSY irreducible representation includes, in addition to Yang–Mills fields [Formula: see text], [Formula: see text] ghost odd-valued fields [Formula: see text], [Formula: see text] and [Formula: see text] even-valued [Formula: see text] for [Formula: see text], [Formula: see text]. To construct the quantum action, [Formula: see text], by adding to the classical action, [Formula: see text], of an [Formula: see text]-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires [Formula: see text] additional fields, [Formula: see text]: antighost [Formula: see text], [Formula: see text] even-valued [Formula: see text], 3 odd-valued [Formula: see text] and Nakanishi–Lautrup [Formula: see text] fields. The action of [Formula: see text] transformations on new fields as [Formula: see text]-irreducible representation space is realized. These transformations are the [Formula: see text] BRST symmetry transformations for the vacuum functional, [Formula: see text]. The space of all fields [Formula: see text] proves to be the space of an irreducible representation of the fields [Formula: see text] for [Formula: see text]-parametric SUSY transformations, which contains, in addition to [Formula: see text] the [Formula: see text] ghost–antighost, [Formula: see text], even-valued, [Formula: see text], odd-valued [Formula: see text] and [Formula: see text] fields. The quantum action is constructed by adding to [Formula: see text] an [Formula: see text]-exact gauge-fixing term with a gauge boson, [Formula: see text]. The [Formula: see text] SUSY transformations are by [Formula: see text] BRST transformations for the vacuum functional, [Formula: see text]. The procedures are valid for any admissible gauge. The equivalence with [Formula: see text] BRST-invariant quantization method is explicitly found. The finite [Formula: see text] BRST transformations are derived and the Jacobians for a change of variables related to them but with field-dependent parameters in the respective path integral are calculated. They imply the presence of a corresponding modified Ward identity related to a new form of the standard Ward identities and describe the problem of a gauge-dependence. An introduction into diagrammatic Feynman techniques for [Formula: see text] BRST invariant quantum actions for Yang–Mills theory is suggested.

scholarly journals Field-dependent BRST–anti-BRST Lagrangian transformations

2015 ◽  
Vol 30 (04n05) ◽  
pp. 1550021 ◽  
Author(s):  
Pavel Yu. Moshin ◽  
Alexander A. Reshetnyak
Keyword(s):  
Symmetry Breaking ◽  
Ward Identity ◽  
Brst Symmetry ◽  
Lagrangian Formalism ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Change Of Variables ◽  
Field Dependent ◽  
Renormalization Group Approach ◽  
Quantum Action

We continue our study of finite BRST–anti-BRST transformations for general gauge theories in Lagrangian formalism, initiated in [arXiv:1405.0790 [hep-th] and arXiv:1406.0179 [hep-th]], with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters, and prove the correctness of the explicit Jacobian in the partition function announced in [arXiv:1406.0179 [hep-th]], which corresponds to a change of variables with functionally dependent parameters λa = UaΛ induced by a finite Bosonic functional Λ(ϕ, π, λ) and by the anticommuting generators Ua of BRST–anti-BRST transformations in the space of fields ϕ and auxiliary variables πa, λ. We obtain a Ward identity depending on the field-dependent parameters λa and study the problem of gauge dependence, including the case of Yang–Mills theories. We examine a formulation with BRST–anti-BRST symmetry breaking terms, additively introduced into the quantum action constructed by the Sp(2)-covariant Lagrangian rules, obtain the Ward identity and investigate the gauge independence of the corresponding generating functional of Green's functions. A formulation with BRST symmetry breaking terms is developed. It is argued that the gauge independence of the above generating functionals is fulfilled in the BRST and BRST–anti-BRST settings. These concepts are applied to the average effective action in Yang–Mills theories within the functional renormalization group approach.

scholarly journals Finite BRST–anti-BRST transformations in generalized Hamiltonian formalism

2014 ◽  
Vol 29 (27) ◽  
pp. 1450159 ◽  
Author(s):  
Pavel Yu. Moshin ◽  
Alexander A. Reshetnyak
Keyword(s):  
Dynamical Systems ◽  
Path Integral ◽  
Hamiltonian Path ◽  
Hamiltonian Formalism ◽  
Lagrangian Formalism ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Field Dependent ◽  
Gauge Fixing ◽  
Constrained Dynamical Systems

We introduce the notion of finite BRST–anti-BRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST–anti-BRST transformations for the Yang–Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790 [hep-th]], special field-dependent BRST–anti-BRST transformations with functionally-dependent parameters λa= ∫ dt(saΛ), generated by a finite even-valued function Λ(t) and by the anticommuting generators saof BRST–anti-BRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters λaand study the problem of gauge dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary Rξ-like gauges in the Yang–Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790 [hep-th]], which justifies the unitarity of the S-matrix in the Lagrangian approach.

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 Superfield approach to nilpotent symmetries in 3D Jackiw–Pi model of massive non-Abelian theory

2014 ◽  
Vol 92 (9) ◽  
pp. 1033-1042 ◽  
Author(s):  
S. Gupta ◽  
R. Kumar ◽  
R.P. Malik
Keyword(s):  
Gauge Theory ◽  
Brst Symmetry ◽  
Point Of View ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Abelian Gauge Theory ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Symmetry Transformations ◽  
Augmented Superfield Formalism

In the available literature, only the Becchi–Rouet–Stora–Tyutin (BRST) symmetries are known for the Jackiw–Pi model of the three (2 + 1)-dimensional (3D) massive non-Abelian gauge theory. We derive the off-shell nilpotent [Formula: see text] and absolutely anticommuting (sbsab + sabsb = 0) (anti-)BRST transformations s(a)b corresponding to the usual Yang–Mills gauge transformations of this model by exploiting the “augmented” superfield formalism where the horizontality condition and gauge invariant restrictions blend together in a meaningful manner. There is a non-Yang–Mills (NYM) symmetry in this theory, too. However, we do not touch the NYM symmetry in our present endeavor. This superfield formalism leads to the derivation of an (anti-)BRST invariant Curci–Ferrari restriction, which plays a key role in the proof of absolute anticommutativity of s(a)b. The derivation of the proper anti-BRST symmetry transformations is important from the point of view of geometrical objects called gerbes. A novel feature of our present investigation is the derivation of the (anti-)BRST transformations for the auxiliary field ρ from our superfield formalism, which is neither generated by the (anti-)BRST charges nor obtained from the requirements of nilpotency and (or) absolute anticommutativity of the (anti-)BRST symmetries for our present 3D non-Abelian 1-form gauge theory.

Quantum gauge theories

2021 ◽  
pp. 223-268
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Background Field ◽  
Field Method ◽  
Extensive Discussion ◽  
Functional Integrals ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Background Field Method ◽  
Popov Method

This chapter, which is the last chapter in Part I, is devoted to an extensive discussion of quantum gauge theories, which is based on functional integrals and Lagrangian quantization. After introducing the notion of a Yang-Mills gauge theory, the Faddeev-Popov method (also known as the DeWitt-Faddeev-Popov procedure) is explained. Starting from this point, the BRST symmetry is formulated, and the corresponding Ward identities (called Slavnov-Taylor identities in some cases) established. More specialized subjects, such as the gauge dependence of effective action and the background field method, are dealt with in detail. In addition, Yang-Mills theory is analyzed as a primary example of general theorems concerning the renormalization of gauge theories.

THE sp(3) BRST HAMILTONIAN FORMALISM FOR THE YANG–MILLS FIELDS

2008 ◽  
Vol 23 (10) ◽  
pp. 737-750 ◽  
Author(s):  
CARMEN IONESCU
Keyword(s):  
Brst Symmetry ◽  
Hamiltonian Formalism ◽  
General Rule ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Yang Mills ◽  
First Approximation ◽  
Gauge Term ◽  
Generalized Symmetry ◽  
Gauge Fixing

The paper presents in all its nontrivial details the sp(3) BRST Hamiltonian formalism. It is based on structuring the extended phase space on many levels. In this picture, the standard BRST symmetry appears as being only the first approximation of a generalized symmetry, acting as a horizontal (same level) operator. The gauge-fixing problem is completely solved by formulating a theorem and a general rule which allow the choice of a simple gauge term. As an example, the Hamiltonian sp(3) quantization of the Yang–Mills model is exhaustively presented.

scholarly journals CONSISTENT AXIAL-LIKE GAUGE FIXING ON HYPERTORI

1994 ◽  
Vol 09 (31) ◽  
pp. 2913-2926 ◽  
Author(s):  
EDWIN LANGMANN ◽  
MANFRED SALMHOFER ◽  
ALEX KOVNER
Keyword(s):  
Gauge Group ◽  
Discrete Group ◽  
Gauge Condition ◽  
Polyakov Loop ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Axial Gauge ◽  
Gauge Fixing ◽  
Gribov Copies

We analyze the Gribov problem for SU (N) and U (N) Yang–Mills fields on d-dimensional tori, d = 2, 3, …. We give an improved version of the axial gauge condition and find an infinite, discrete group [Formula: see text] where r = N − 1 and N for G = SU (N) and U (N) respectively, containing all gauge transformations compatible with that condition. This residual gauge group [Formula: see text] provides all Gribov copies for nondegenerate configurations in d = 2 and for those of them for which all winding numbers of the Wilson–Polyakov loop in one direction vanish in d ≥ 3. This shows that the space of gauge orbits is an orbifold. We derive this result both in the Lagrangian and in the Hamiltonian framework.

scholarly journals GAUGE MODELS IN MODIFIED TRIPLECTIC QUANTIZATION

2001 ◽  
Vol 16 (26) ◽  
pp. 4297-4319 ◽  
Author(s):  
B. GEYER ◽  
P. M. LAVROV ◽  
P. YU. MOSHIN
Keyword(s):  
2D Gravity ◽  
Brst Symmetry ◽  
Antisymmetric Tensor ◽  
Explicit Solutions ◽  
Tensor Fields ◽  
Gauge Models ◽  
Gauge Fixing ◽  
Quantum Action

The modified triplectic quantization is applied to several well-known gauge models: the Freedman–Townsend model of non-Abelian antisymmetric tensor fields, W2 gravity, and 2D gravity with dynamical torsion. For these models we obtain explicit solutions of those generating equations that determine the quantum action and the gauge-fixing functional. Using these solutions, we construct the vacuum functional, determine the Sp(2)-invariant effective actions and obtain the corresponding transformations of extended BRST symmetry.

scholarly journals SOFT BREAKING OF BRST SYMMETRY AND GAUGE DEPENDENCE

2012 ◽  
Vol 27 (13) ◽  
pp. 1250067 ◽  
Author(s):  
P. M. LAVROV ◽  
O. V. RADCHENKO ◽  
A. A. RESHETNYAK
Keyword(s):  
Green's Functions ◽  
Gauge Theories ◽  
Green’S Functions ◽  
Brst Symmetry ◽  
Gauge Dependence ◽  
Gauge Fixing ◽  
Definition Of ◽  
Continue Investigation ◽  
Arbitrary Gauge ◽  
General Gauge

We continue investigation of soft breaking of BRST symmetry in the Batalin–Vilkovisky (BV) formalism beyond regularizations like dimensional ones used in our previous paper [JHEP 1110, 043 (2011)]. We generalize a definition of soft breaking of BRST symmetry valid for general gauge theories and arbitrary gauge fixing. The gauge dependence of generating functionals of Green's functions is investigated. It is proved that such introduction of a soft breaking of BRST symmetry into gauge theories leads to inconsistency of the conventional BV formalism.

scholarly journals QUANTUM AND CLASSICAL GAUGE SYMMETRIES IN A MODIFIED QUANTIZATION SCHEME

2001 ◽  
Vol 16 (10) ◽  
pp. 1775-1788 ◽  
Author(s):  
KAZUO FUJIKAWA ◽  
HIROAKI TERASHIMA
Keyword(s):  
Gauge Symmetry ◽  
Brst Symmetry ◽  
Mass Term ◽  
Large Mass ◽  
Quantization Scheme ◽  
Mass Limit ◽  
Classical Action ◽  
Gauge Invariant ◽  
Gauge Fixing ◽  
Nonlinear Gauge

The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the nonlinear gauge [Formula: see text] of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional local Faddeev–Popov formula without taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive nongauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge-invariant, has a more intrinsic meaning. It is suggested that we extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry.

Sign in / Sign up

Export Citation Format

Share Document