Quantization of Gauge Systems

Name: Quantization of Gauge Systems
Brand: Princeton University Press
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Marc Henneaux Claudio Teitelboim(Autor*in)

Princeton University Press

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Erschienen am 16. Juni 2020

552 Seiten

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Inhalt

Cover Page
Half-title Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Notations
Chapter One: Constrained Hamiltonian Systems
1.1. Gauge Invariance-Constraints
1.1.1. The Lagrangian as a Starting Point: Primary Constraints
1.1.2. Conditions on the Constraint Functions
1.1.3. The Canonical Hamiltonian
1.1.4. Action Principle in Hamiltonian Form
1.1.5. Secondary Constraints
1.1.6. Weak and Strong Equations
1.1.7. Restrictions on the Lagrange Multipliers
1.1.8. Irreducible and Reducible Cases
1.1.9. Total Hamiltonian
1.1.10. First-Class and Second-Class Functions
1.2. First-Class Constraints as Generators of Gauge Transformations
1.2.1. Transformations That Do Not Change the Physical State. Gauge Transformations
1.2.2. A Counterexample to the Dirac Conjecture
1.2.3. The Extended Hamiltonian
1.2.4. Extended Action Principle
1.3. Second-Class Constraints: The Dirac Bracket
1.3.1. Separation of First-Class and Second-Class Constraints
1.3.2. Treatment of Second-Class Constraints: An Example
1.3.3. Dirac Bracket
1.3.4. Reducible First-Class and Second-Class Constraints
1.4. Gauge Fixation-Independent Degrees of Freedom
1.4.1. Canonical Gauges
1.4.2. Counting of Degrees of Freedom
1.4.3. Do All Second-Class Constraints Arise from Gauge Fixation?
1.5. Gauge-Invariant Functions
1.5.1. Functions on the Constraint Surface
1.5.2. Classical Observables
1.5.3. Algebraic Characterization of the Observables
1.5.4. Gauge-Invariant Extensions
1.6. Examples
1.6.1. System with n Generations of Constraints
1.6.2. L = 0
1.6.3. More on the Consistency Algorithm
Appendix l.A. Global proof of G = 0
Exercises
Chapter Two: Geometry of the Constraint Surface
2.1. Induced Two-Form on the Constraint Surface
2.1.1. An Analogy: Surfaces in Minkowski Space
2.1.2. Geometry of Phase Space (Symplectic Geometry)
2.1.3. Induced Two-Form
2.2. First-Class Constraint Surface
2.2.1. Rank of Induced Two-Form
2.2.2. Null Surfaces and Gauge Orbits
2.2.3. Reduced Phase Space
2.3. Second-Class Constraints
2.3.1. Rank of Induced Two-Form
2.3.2. Dirac Bracket Revisited
2.3.3. Solving the Constraints inside the Action
2.4. Mixed Case
Appendix 2.A. More on the Structure of the Reduced Phase Space
Exercises
Chapter Three: Gauge Invariance of the Action
3.1. Structure of the Gauge Symmetries
3.1.1. Notations
3.1.2. Gauge Transformations
3.1.3. Noether Identities
3.1.4. Gauge Group-Gauge Algebra
3.1.5. Trivial Gauge Transformations
3.1.6. Independent Noether Identities
3.1.7. Generating Sets
3.1.8. "Open Algebras
3.1.9. Reducible Generating Sets
3.1.10. Relation between Different Generating Sets
3.1.11. Generating Sets and Gauge Orbits
3.2. Gauge Transformations of the Extended Action
3.2.1. Algebra of the Constraints
3.2.2. Gauge Transformations
3.2.3. Another Generating Set
3.2.4. Gauge Transformations as Canonical Transformations
3.2.5. Open and Closed Algebras
3.2.6. Reducible First-Class Constraints
3.2.7. Conclusions
3.3. Gauge Transformations of the Original Lagrangian Action
3.3.1. Gauge Symmetries of ST and SL
3.3.2. Proof of the Dirac Conjecture under Simplifying Assumptions
3.3.3. Lagrangian Form of the Gauge Transformations-Basic Equations
3.3.4. Solution of the Basic Equations
3.3.5. Lagrange Multiplier Dependence of Gauge Transformations
3.3.6. Gauge Invariance and Degree of Freedom Count
3.3.7. Total and Extended Hamiltonians Compared and Contrasted
3.4. Noncanonical Gauges
3.4.1. Derivative Gauges
3.4.2. Multiplier Gauges
3.4.3. Reducible Gauge Transformations-Redundant Gauge Conditions
Exercises
Chapter Four: Generally Covariant Systems
4.1. Introduction
4.2. Time as a Canonical Variable-Zero Hamiltonian
4.2.1. Parametrized Systems
4.2.2. Zero Hamiltonian
4.2.3. Parametrization and Explicit Time Dependence
4.3. Time Reparametrization Invariance
4.3.1. Form of Gauge Transformations
4.3.2. Must the Hamiltonian Be Zero for a Generally Covariant System?
4.3.3. Simple Example of a Generally Covariant System with a Nonzero Hamiltonian
4.4. "True Dynamics" versus Gauge Transformations
4.4.1. Interpretation of the Formalism
4.4.2. Reduced Phase Space
Exercises
Chapter Five: First-Class Constraints: Further Developments
5.1. Preliminaries and Notations
5.2. Abelianization of Constraints
5.2.1. Ambiguity in the Description of the Constraint Surface
5.2.2. Abelianization Theorem
5.3. Exterior Derivative Operator along the Gauge Orbits("Longitudinal Derivative")
5.3.1. Definition of Longitudinal Derivative
5.3.2. Longitudinal Cohomology
5.3.3. Representation of Longitudinal Derivative in the Irreducible Case
5.3.4. Representation of Longitudinal Derivative in the Reducible Case
5.3.5. Phase Space Characterization of Longitudinal Forms
5.4. Hamilton-Jacobi Theory
5.4.1. Unconstrained Systems-Complete Integrals
5.4.2. Unconstrained Systems-Incomplete Integrals
5.4.3. Constrained Systems
5.4.4. Gauge Invariance of the Hamilton-Jacobi Solutions
5.4.5. Hamilton Principal Function
Exercises
Chapter Six: Fermi Degrees of Freedom: Classical Mechanics over a Grassmann Algebra
6.1. Fermions and Anticommuting c-Numbers
6.2. Formal Properties of Anticommuting c-Numbers
6.2.1. Grassmann Algebra
6.2.2. Superfunctions
6.2.3. Grassmann parity
6.2.4. Complex Conjugation
6.3. Changes of Variables
6.3.1. Invertible Matrices
6.3.2. Invertible Changes of Variables
6.4. Canonical Formalism in the Presence of Odd Variables
6.5. Generalized Poisson Bracket
6.5.1. Definition
6.5.2. Properties of the Generalized Poisson brackets
6.5.3. Algebra of Superfunctions over Phase Space as the Central Object in Grassmann Mechanics
6.6. Physical Fermions Need First-Order Equations
6.6.1. A Simple Model System
6.6.2. Negative Norm States Generic for Nondegenerate Fermionic Lagrangians
6.6.3. Supersymmetry
6.7. Geometry of Phase Space in the Anticommuting Case
6.7.1. Phase Space
6.7.2. Supersymplectic Geometry
Exercises
Chapter Seven: Constrained Systems with Fermi Variables
7.1. Odd-Dimensional Phase Space
7.1.1. Example
7.1.2. Boundary Term in Action Principle
7.1.3. Alternative Boundary Conditions in the Hamiltonian Variational Principle for Bosonic Variables
7.2. Incorporation of Appropriate Sign Factors
7.2.1. Gauge Transformations
7.2.2. Gauge Orbits-Exterior Derivative along the Gauge Orbits
Exercises
Chapter Eight: Graded Differential Algebras-Algebraic Structure of the BRST Symmetry
8.1. Introduction-Ghosts
8.2. Graded Differential Algebras
8.2.1. Supercommutative Algebras
8.2.2. Examples
8.2.3. Graded Lie Algebra of Graded Derivations
8.2.4. Gradings
8.2.5. Ideals
8.2.6. Differentials-Cohomology Algebras
8.2.7. Contracting Homotopy
8.2.8. Cohomology for the Lie Algebra of Derivations
8.2.9. Differential modulo
8.3. Resolution
8.3.1. Definition
8.3.2. Example
8.4. Elements of Homological Perturbation Theory
8.4.1. Main Theorem
8.4.2. Proof of the Main Theorem: (i) Existence of s
8.4.3. Proof of the Main Theorem: (ii) Evaluation of Hk(s)
8.4.4. Comments
8.5. Geometric Application: The BRST Construction in Brief
8.5.1. Introduction
8.5.2. Geometric Ingredients
8.5.3. BRST Differential
8.5.4. Canonical Action of s
Exercises
Chapter Nine: BRST Construction in the Irreducible Case
9.1. Koszul-Tate Resolution
9.1.1. Definition
9.1.2. Homology of 6
9.2. Extended Phase Space
9.2.1. Ghosts and Longitudinal d
9.2.2. Bracket Structure-Ghost Number
9.2.3. 8 and d in the Extended Phase Space
9.3. Bringing 6 and d Together: The BRST Symmetry as a Canonical Transformation
9.3.1. BRST Generator
9.3.2. Existence of the BRST Generator
9.3.3. The BRST Generator Is Unique up to Canonical Transformations
9.4. The BRST Generator in Simple Cases-Rank
9.4.1. Abelian Constraints
9.4.2. Constraints that Close According to a Group
9.4.3. Higher Order Structure Functions
9.4.4. Rank
9.5. Conclusions
Appendix 9.A. Proof of Theorem 9.1 (Homology of 6)
9.A.1. 6-Covering of Phase Space
9.A.2. Homology of 6 on O, at Positive Antighost Number
9.A.3. Homology of 6 on Va at Positive Antighost Number
9.A.4. Homology of 8
Exercises
Chapter Ten: BRST Construction in the Reducible Case
10.1. The Simplest Example
10.2. Description of Reducible Theories
10.2.1. First-Order Reducibility Functions
10.2.2. Completeness in Terms of Strong Equalities
10.2.3. Higher Order Reducibility Functions
10.2.4. Ambiguity in the Reducibility Functions
10.2.5. Canonical Form
10.3. The Koszul-Tate Differential
10.3.1. Nontrivial Cycles and How to Kill Them
10.3.2. Homology of
10.4. More on the Longitudinal Exterior Differential
10.4.1. Problem with the Definition of the Extended Phase Space
10.4.2. The Longitudinal Differential
10.4.3. Auxiliary Differential
10.4.4. Auxiliary Grading
10.4.5. The Differential D
10.4.6. Cohomology of D
10.4.7. Conclusions
10.5. BRST Transformation
10.5.1. Extended Phase Space
10.5.2. Combining with D
10.5.3. Equations Determining the BRST Generator
10.5.4. Existence of the BRST Generator
10.5.5. Uniqueness of the BRST Generator
10.6. Conclusions
Appendix 10.A. Proofs of Theorems 10.1 through 10.4
10.A.1. Covering of Phase Space
10.A.2. Proof of Theorem 10.1
10. A.3. Proofs of Theorems 10.2 and 10.3
10.A.4. Proof of Theorem 10.4
Exercises
Chapter Eleven: Dynamics of the Ghosts-Gauge-Fixed Action
11.1. BRST Cohomology and the Poisson Bracket
11.1.1. BRST Observables
11.1.2. What Is the Meaning of the Higher Cohomological Groups
11.1.3. Ghost Transformation Law under Global Symmetries
11.2. Ghost Dynamics
Gauge Fixing. The BRST Function as the Generator of a Symmetry
11.2.1. BRST-Invariant Hamiltonians
11.2.2. BRST Symmetry-Gauge-Fixed Action
11.2.3. Comments
11.3. Nonminimal Solutions
11.3.1. Nonminimal Sector
11.3.2. The Lagrange Multipliers as Canonical Variables
11.3.3. The Faddeev-Popov Action
11.3.4. Lagrangian Form of the BRST Symmetry-The BRST Generator as a Noether Charge
11.3.5. Hamilton Principal Function and Ghosts
Exercises
Chapter Twelve: The BRST Transformation in Field Theory
12.1. Local Functionals and Nonintegrated Densities
12.2. Local Completeness and Regularity Conditions
12.2.1. Hamiltonian Definition of a Local Gauge Theory
12.2.2. Regularity Conditions
12.2.2a. Local Completeness of the Constraint Functions
12.2.2b. Zero Is a Regular Value of the Map Defined by the Constraint Functions
12.2.3. Local Completeness of the Reducibility Functions
12.3. Locality of the BRST Charge
12.3.1. Homology of S modulo dkjk as the Central Issue in the Problem of the Spacetime Locality of the BRST Formalism
12.3.2. Proof of Theorem 12.5: (i) Local Homology of
12.3.3. Proof of Theorem 12.5: (ii) Homology of modulo
12.3.4. Locality of the Gauge-Fixed Action
Exercises
Chapter Thirteen: Quantum Mechanics of Constrained Systems: Standard Operator Methods
13.1. Quantization of Second-Class Constraints
13.1.1. An Example
13.1.2. Correspondence Rules in the General Case
13.1.3. Difficulties
13.2. Reduced Phase Space Quantization of First-Class Constraints
13.2.1. Description of the Method
13.2.2. Gauge Conditions
13.2.3. Difficulties
13.3. Dirac Quantization of First-Class Constraints
13.3.1. Formal Aspects
13.3.2. Anomalies
13.3.3. Generally Covariant Systems
13.3.4. Scalar Product
13.3.5. A Different Derivation of the Physical Condition
13.3.6. Projected Kernel of Gauge-Invariant Operators
13.4. Dirac-Fock Quantization of First-Class Constraints
13.4.1. Definition
13.4.2. Physical Subspace
13.4.3. Conclusions
Exercises
Chapter Fourteen: BRST Operator Method-Quantum BRST Cohomology
14.1. General Features
14.1.1. States and Operators
14.1.2. Ghost Number
14.1.3. Physical State Condition
14.1.4. Quantum BRST Cohomology
14.1.5. Anomalies
14.2. Analysis of Quantum BRST Cohomology: General Theorems
14.2.1. Jordan Canonical Form of the BRST Charge: Operator Cohomology versus State Cohomology
14.2.1a. State Cohomology
14.2.1b. Operator Cohomology
14.2.1c. Lefschetz Trace Formula
14.2.2. Duality Formula for the Operator Cohomology
14.2.3. (Pseudo-)Unitary Representations of the BRST-Ghost Number Algebra
14.2.4. Duality Formula for the State Cohomology
14.2.5. Physical States and Ghost Number
14.2.6. No Negative Norm State Criterion
14.3. Time Evolution
14.3.1. Schrodinger Equation
14.3.2. Unitarity in the Physical Subspace
14.4. BRST Quantization in the Fock Representation
14.4.1. BRST Charge and Ghost Number Operator
14.4.2. Quartet Mechanism
14.4.3. Comments
14.5. BRST Quantization and Solutions of the Constraint Equations
14.5.1. Quantum Constraints and Ordering of
14.5.2. Redefinitions of the Constraints
14.5.3. BRST Cohomology at Ghost Number
14.5.4. Forming Ghost Number Zero States
14.5.5. BRST Formalism and Projected Kernels 323 Exercises
Exercises
Chapter Fifteen: Path Integral for Unconstrained Systems
15.1. Path Integral Method of Bose Systems-Basic Features
15.1.1. Path Integral as a Kernel
15.1.2. Comments
15.1.3. Quantum Averages of Functionals
15.1.4. Equations of Motion-Schwinger-Dyson Equations
15.1.5. Stationary Phase Method-Lagrangian Path Integral
15.2. Path Integral in the Holomorphic Representation (Bose Systems)
15.2.1. Definition of Holomorphic Representation
15.2.2. Path Integral
15.3. Path Integral for Systems with Indefinite Metric
15.3.1. Introduction
15.3.2. Coordinate Representation
15.3.3. Path Integral in the Coordinate Representation
15.3.4. Holomorphic Representation
15.3.5. Path Integral in the Holomorphic Representation
15.4. Path Integral for Fermions
15.4.1. Path Integral in the Holomorphic Representation
15.4.2. Path Integral for the Weyl Symbol of the Evolution Operator
15.4.2a. Action Principle
15.4.2b. Weyl Correspondence Rule
15.4.2c. Path Integral Representation of the Evolution Operator
15.4.3. Example: Spin-i in a Magnetic Field
15.4.4. Ghost Transition Amplitude
15.5. A First Bite at the Antifield Formalism
15.5.1. Koszul-Tate Differential Associated with the Stationary Surface
15.5.2. Antibracket
15.5.3. Schwinger-Dyson Operator
15.5.4. Geometric Interpretation of A and of the Antibracket
15.5.5. The Antibracket Does Not Define a Measure
Exercises
Chapter Sixteen: Path Integral for Constrained Systems
16.1. Path Integral for Second-Class Constraints
16.1.1. Derivation of the Path Integral
16.1.2. Difficulties
16.2. Reduced Phase Space Path Integral
16.2.1. Derivation of the Path Integral
16.2.2. Faddeev Formula
16.2.3. Gauge Independence of Path Integral for a Parametrized System Illustrated. Equivalence of the Gauges t = r and t = 0
16.2.3a. Reduced Phase Space Transition Amplitude as a Reduced Phase Space Path Integral
16.2.3b. Canonical Gauge Conditions
16.2.3c. Gauge t = 0
16.2.3d. Gauge
16.3. BRST Path Integral in the Fock Representation
16.3.1. Construction
16.3.2. Example
16.4. Fradkin-Vilkovisky Theorem-Ward Identities
16.4.1. Theorem
16.4.2. Quantum Averages and BRST Cohomological Classes
16.4.3. Ward Identities
16.4.4. Zinn-Justin Equation
16.5. BRST Path Integral in the Schrodinger Representation
16.5.1. Projected Kernel of the Evolution Operator
16.5.2. Semiclassical Approximation
16.5.3. Composition Rule
16.5.4. Comparison with Reduced Phase Space Path Integral
16.5.5. BRST Path Integral for Generally Covariant Systems-Proper Time Gauge-Causal Propagator
16.5.6. Path Integral in Multiplier Gauges
Exercises
Chapter Seventeen: Antifleld Formalism: Classical Theory
17.1. Covariant Phase Space
17.1.1. Path Integral and Spacetime Covariance
17.1.2. Covariant Phase Space in the Absence of Gauge Invariance
17.1.3. Covariant Phase Space in the Presence of Gauge Freedom
17.1.4. Lagrangian Homological Perturbation Theory
17.1.5. Regularity Conditions
17.2. Koszul-Tate Resolution and Longitudinal
17.2.1. Koszul-Tate Resolution
17.2.2. Any Gauge Transformation that Vanishes On-Shell Is a Trivial Gauge Transformation
17.2.3. Longitudinal Exterior Differential
17.2.4. 6 and Spacetime Locality
17.3. BRST Symmetry-Master Equation
17.3.1. Antibracket Structure
17.3.2. Master Equation
17.3.3. Solution of the Master Equation
17.3.4. Canonical Transformation in the Antibracket
17.3.5. Nonrninimal Solutions
17.3.6. Antibracket and BRST Cohomology
17.4. Gauge Invariance of the Solution of the Master Equation
17.4.1. Abelian Form of S
17.4.2. Gauge Transformations of S
Exercises
Chapter Eighteen: Antifield Formalism and Path Integral
18.1. Quantum Master Equation
18.1.1. Integration of p-Vectors on a Supermanifold
18.1.2. Invariance under Canonical "Phase" Transformations
18.1.3. Derivation of Quantum Master Equation
18.1.4. Quantum Averages
18.1.5. Quantum BRST Symmetry-Ward Identity
18.1.6. Zinn-Justin Equation
18.2. Solution of the Quantum Master Equation
18.2.1. Ambiguity
18.2.2. Ambiguity
18.2.3. Example
18.2.4. Dimensional Regularization
18.3. Invariance of the Formalism under Canonical Transformations in the Antibracket
18.3.1. Antifield Formalism Can Only Be Justified up to Quantum Ambiguities in the Measure
18.3.2. More on Canonical Transformations
18.3.3. Transformation of
18.3.4. Invariance of the Path Integral
18.3.5. The Path Integral in the Abelian Representation
18.4. Equivalence of Antifield and Hamiltonian Formalisms
18.4.1. Gauge-Fixed Form of the BRST Symmetry in the Antifield Formalism
18.4.2. Digression. Gauge-Fixed BRST Cohomology
18.4.3. Equivalence of Antifield BRST Symmetry and Hamiltonian BRST Symmetry
18.4.4. The Antifield Path Integral Based on SL, ST, and SE Are the Same
18.4.5. Antifield Formalism for the Extended Hamiltonian Action
Exercises
Chapter Nineteen: Free Maxwell Theory. Abelian Two-Form Gauge Field
19.1. Free Maxwell Field
19.1.1. Hamiltonian Analysis
19.1.2. Classical BRST Cohomology
19.1.3. Antifield Formalism
19.1.4. Path Integral-Gauge-Fixed Action
19.1.4a. Hamiltonian Treatment
19.1.4b. Antifield Treatment
19.1.5. Faddeev-Popov Determinant
19.1.6. Operator Quantization
19.1.7. Gauge
19.1.7a. Antifield Treatment
19.1.7b. Hamiltonian Treatment
19.1.8. Temporal Gauge
19.2. Abelian 2-Form Gauge Fields
19.2.1. Hamiltonian Analysis
19.2.2. Classical BRST Cohomology
19.2.3. Nonminimal Sector-Operator Formalism
19.2.4. Generalization: Hamiltonian Nonminimal Sector for Arbitrary Reducible Theories
19.2.5. Path Integral
19.2.6. Generalization: Antifield Nonminimal Sector for Arbitrary Reducible Theories
Exercises
Chapter Twenty: Complementary Material
20.1. Exterior Calculus on a Supermanifold: Conventions
20.2. Integration on a Supermanifold
20.2.1. Definition
20.2.2. Supertrace-Superdeterminant
20.2.3. Change of Variables. Superdensities
20.2.4. Delta Function-Gaussian Integrals
20.2.5. Liouville Measure
20.3. Quantization of Fermi Degrees of Freedom: Clifford Algebras
20.3.1. Introduction
20.3.2. Clifford Algebras with an Even Number of Generators
20.3.2a. Clifford Algebra Associated with (20.37)
20.3.2b. Clifford Algebra Associated with (20.38)
20.3.2c. Clifford Algebra Associated with (20.39)
20.3.2d. Combining the Representations of (20.37)-(20.39)
20.3.2e. Grassmann Parity
20.3.3. Clifford Algebra with an Odd Number of Generators
20.3.3a. Irreducible Representations of the Clifford Algebra
20.3.3b. Reality Conditions
Exercises
Bibliography
Index

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