- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
St And Brs Symmetries, Stochastic Field Equations
St And Brs Symmetries, Stochastic Field Equations
- Chapter:
- (p.396) 16 ST AND BRS SYMMETRIES, STOCHASTIC FIELD EQUATIONS
- Source:
- Quantum Field Theory and Critical Phenomena
- Author(s):
JEAN ZINN-JUSTIN
- Publisher:
- Oxford University Press
Section 15.3 introduced a transformation depending on anticommuting parameters, to prove the geometric stability of homogeneous spaces under renormalization. There is a set of topics, stochastic field equations, gauge theories, in which similar transformations are met. These new problems have one common feature: they all involve a constraint equation to which, by a set of formal transformations, is associated a quantum action. This action has an anticommuting type symmetry which has no geometric origin but is merely a consequence of these transformations. This chapter first discusses this mathematical structure from a rather formal point of view, using a notation adapted to a finite number of degrees of freedom. It explains the appearance of Slavnov–Taylor symmetry, which is a conventional non-linear symmetry, in the integral representation of constraint equations. It then shows how it leads to a symmetry with anticommuting parameters first discovered in quantized gauge theories by Becchi, Rouet and Stora and, therefore, called BRS symmetry. Its generator has a vanishing square, and generalizes exterior differentiation. This symmetry is remarkably stable against a number of algebraic deformations and this explains its role in the context of stochastic equations. In some cases it can be expressed in compact form by introducing Grassmann coordinates. The chapter shows how BRS symmetry can encode the compatibility conditions of a system of linear first order differential equations. It exhibits the special form BRS symmetry takes when the constraint equations apply to group manifolds.
Keywords: Slavnov–Taylor symmetry, Grassmann coordinates, stochastic field equations, group manifolds
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- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
