scholarly journals EMERGENCE OF THE HAAR MEASURE IN THE STANDARD FUNCTIONAL INTEGRAL REPRESENTATION OF THE YANG-MILLS PARTITION FUNCTION

1996 ◽  
Vol 11 (30) ◽  
pp. 2451-2461 ◽  
Author(s):  
H. REINHARDT
Keyword(s):  
Partition Function ◽  
Path Integral ◽  
Haar Measure ◽  
Transition Amplitude ◽  
Functional Integral ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Gauge Fixing ◽  
Popov Method

The conventional path integral expression for the Yang–Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the nonperturbative regime. We show, however, that it yields the gauge-invariant partition function where the projection onto gauge-invariant wave functions is explicitly performed by integrating over the compact gauge group. In a variant of maximal Abelian gauge the Haar measure arises in the conventional Yang-Mills path integral from the Faddeev-Popov determinant.

scholarly journals FROM PEIERLS BRACKETS TO A GENERALIZED MOYAL BRACKET FOR TYPE-I GAUGE THEORIES

2007 ◽  
Vol 04 (03) ◽  
pp. 349-360 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
COSIMO STORNAIOLO
Keyword(s):  
Vector Fields ◽  
Field Operator ◽  
Functional Integral ◽  
Gauge Fields ◽  
Order Differential Operator ◽  
Type I ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Group Invariant ◽  
Gauge Fixing

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang–Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to iħ times the Peierls bracket to lowest order in ħ.

scholarly journals SIGNIFICANCE OF ZERO MODES IN PATH INTEGRAL QUANTIZATION OF SOLITONIC THEORIES WITH BRST INVARIANCE

1996 ◽  
Vol 11 (28) ◽  
pp. 4985-4998
Author(s):  
J.-G. ZHOU ◽  
F. ZIMMERSCHIED ◽  
H. J. W. MÜLLER-KIRSTEN ◽  
J.-Q. LIANG ◽  
D. H. TCHRAKIAN
Keyword(s):  
Path Integral ◽  
Transition Amplitude ◽  
Functional Integral ◽  
Field Equations ◽  
Path Integral Quantization ◽  
Zero Modes ◽  
Loop Approximation ◽  
Gauge Fixing ◽  
The One ◽  
Brst Invariance

The significance of zero modes in the path integral quantization of some solitonic models is investigated. In particular a Skyrme-like theory with topological vortices in 1 + 2 dimensions is studied, and with a BRST-invariant gauge fixing a well-defined transition amplitude is obtained in the one-loop approximation. We also present an alternative method which does not necessitate evoking the time dependence in the functional integral, but is equivalent to the original one in dealing with the quantization in the background of the static classical solution of the nonlinear field equations. The considerations given here are particularly useful in — but also limited to — the one-loop approximation.

scholarly journals The Path Integral for (1 + 1)-Dimensional QCD

1997 ◽  
Vol 12 (24) ◽  
pp. 4445-4459 ◽  
Author(s):  
O. Jahn ◽  
T. Kraus ◽  
M. Seeger
Keyword(s):  
Path Integral ◽  
Effective Potential ◽  
Transition Amplitude ◽  
Integral Expression ◽  
Gauge Invariant ◽  
Curved Space ◽  
Gauge Fixing

We derive a path integral expression for the transition amplitude in (1 + 1)-dimensional QCD starting from canonically quantized QCD. Gauge fixing after quantization1 leads to a formulation in terms of gauge invariant but curvilinear variables. Remainders of the curved space are Jacobians, an effective potential, and sign factors just as for the problem of a particle in a box. Based on this result we derive a Faddeev–Popov-like expression for the transition amplitude avoiding standard infinities that are caused by integrations over gauge equivalent configurations.

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.

scholarly journals Gauge Invariance and Mass Gap in (2+1)-Dimensional Yang–Mills Theory

1997 ◽  
Vol 12 (06) ◽  
pp. 1161-1171 ◽  
Author(s):  
Dimitra Karabali ◽  
V. P. Nair
Keyword(s):  
Gauge Invariance ◽  
Gauge Theories ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Mass Gap ◽  
Spatial Dimensions

In terms of a gauge-invariant matrix parametrization of the fields, we give an analysis of how the mass gap could arise in non-Abelian gauge theories in two spatial dimensions.

Anomalies, instantons and axions

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Standard Model ◽  
Path Integral ◽  
Electric Charge ◽  
Gauge Invariant ◽  
Axial Anomaly ◽  
Yang Mills ◽  
Integral Measure ◽  
The Standard Model ◽  
Electroweak Sector

The axial anomaly is derived both from the non-invariance of the path-integral measure under UA(1) transformations and calculations of specific triangle diagrams. It is demonstrated that the anomalous terms are cancelled in the electroweak sector of the standard model, if the electric charge of all fermions adds up to zero. The CP-odd term F̃μν‎Fμν‎ introduced by the axial anomaly is a gauge-invariant renormalisable interaction which is also generated by instanton transitions between Yang–Mills vacua with different winding numbers. The Peceei–Quinn symmetry is discussed as a possible explanation why this term does not contribute to the QCD action.

scholarly journals GAUGE-INVARIANT SOFT MODES IN YANG–MILLS THEORY

2007 ◽  
Vol 16 (09) ◽  
pp. 2789-2793 ◽  
Author(s):  
HILMAR FORKEL
Keyword(s):  
Degrees Of Freedom ◽  
Transition Amplitude ◽  
Collective Excitations ◽  
Analytical Treatment ◽  
Invariant Representation ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Soft Modes ◽  
Dominant Sets ◽  
Vacuum Fields

A gauge-invariant saddle point expansion for the Yang–Mills vacuum transition amplitude on the basis of the squeezed approximation to the vacuum wave functional is outlined. This framework allows the identification of gauge-invariant infrared degrees of freedom which arise as dominant sets of gauge field orbits and provide the principal input for an essentially analytical treatment of soft amplitudes. The analysis of the soft modes sheds new light on how vacuum fields organize themselves into collective excitations and yields a gauge-invariant representation of instanton and meron effects as well as a new physical interpretation for Faddeev–Niemi knots.

scholarly journals Physical quantities and arbitrariness in resolving quantum master equation

2017 ◽  
Vol 32 (11) ◽  
pp. 1750068 ◽  
Author(s):  
Igor A. Batalin ◽  
Peter M. Lavrov
Keyword(s):  
Master Equation ◽  
Path Integral ◽  
Quantum Master Equation ◽  
Abelian Gauge ◽  
Quantization Procedure ◽  
Gauge Fixing ◽  
Physical Quantities

By proceeding with the idea that the presence of physical (BRST invariant) extra factors in the path integral is equivalent to taking into account explicitly the arbitrariness in resolving the quantum master equation, we consider the field–antifield quantization procedure both with the Abelian and the non-Abelian gauge fixing.

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