Classical and Quantum Dynamics of Constrained Hamiltonian Systems Book

1 Introduction 1

2 Singular Lagrangians and Local Symmetries 6

2.1 Introduction 6

2.2 Singular Lagrangians 7

2.3 Algorithm for detecting local symmetries on Lagrangian level 9

2.4 Examples 14

2.5 Generator of gauge transformations and Noether identities 20

3 Hamiltonian Approach. The Dirac Formalism 24

3.1 Introduction 24

3.2 Primary constraints 24

3.3 The Hamilton equations of motion 27

3.3.1 Streamlining the Hamilton equations of motion 29

3.3.2 Alternative derivation of the Hamilton equations 33

3.3.3 Examples 37

3.4 Interative procedure for generating the constraints 43

3.4.1 Particular algorithm for generating the constraints 44

3.5 First and second class constraints. Dirac brackets 46

4 Symplectic Approach to Constrained Systems 51

4.1 Introduction 51

4.2 The case fab singular 54

4.2.1 Example: particle on a hypersphere 58

4.3 Interpretation of W^(L) and F 60

4.4 The Faddeev-Jackiw reduction 62

5 Local Symmetries within the Dirac Formalism 67

5.1 Introduction 67

5.2 Local symmetries and canonical transformations 68

5.3 Local symmetries of the Hamilton equations of motion 70

5.4 Local symmetries of the total and extended action 72

5.5 Local symmetries of the Lagrangian action 75

5.6 Solution of the recursive relations 78

5.7 Reparametrization invariant approach 83

6 The Dirac Conjecture 90

6.1 Introduction 90

6.2 Gauge identities and Dirac's conjecture 90

6.3 General system with two primaries and one secondary constraint 98

6.4 Counterexamples to Dirac's conjecture? 101

7 BFT Embedding of Second Class Systems 108

7.1 Introduction 108

7.2 Summary of the BFT-procedure 109

7.3 The BFT construction 113

7.4 Examples of BFT embedding 116

7.4.1 The multidimensional rotator 116

7.4.2 The Abelian self-dual model 118

7.4.3 Abelian self-dual model and Maxwell-Chern-Simons theory 121

7.4.4 The non-abelian SD model 126

8 Hamilton-Jacobi Theory of Constrained Systems 132

8.1 Introduction 132

8.1.1 Carathéodory's integrability conditions 133

8.1.2 Characteristic curves of the HJ-equations 135

8.2 HJ equations for first class systems 137

8.3 HJ equations for second class systems 139

8.3.1 HPF for reduced second class systems 139

8.3.2 Examples 141

8.3.3 HJ equations for second class systems via BFT embedding 145

8.3.4 Examples 148

9 Operator Quantization of Second Class Systems 154

9.1 Introduction 154

9.2 Systems with only second class constraints 155

9.3 Systems with first and second class constraints 156

9.3.1 Example: the free Maxwell field in the Coulomb gauge 160

9.3.2 Concluding remark 162

10 Functional Quantization of Second Class Systems 164

10.1 Introduction 164

10.2 Partition function for second class systems 165

11 Dynamical Gauges. BFV Functional Quantization 174

11.1 Introduction 174

11.2 Grassmann variables 175

11.3 BFV quantization of a quantum mechanical model 181

11.3.1 The gauge-fixed effective Lagrangian 182

11.3.2 The conserved BRST charge in configuration space 186

11.3.3 The gauge fixed effective Hamiltonian 187

11.3.4 The BRST charge in phase space 190

11.4 Quantization of Yang-Mills theory in the Lorentz gauge 195

11.5 Axiomatic BRST approach 204

11.5.1 The BRST charge and Hamiltonian for rank one theories 205

11.5.2 FV Principal Theorem 209

11.5.3 A large class of gauges 211

11.5.4 Connecting ZΨ with the quantum partition function in a physical gauge. The SU (N) Yang-Mills theory 212

11.6 Equivalence of the SD and MCS models 215

11.7 The physical Hilbert space. Some remarks 221

12 Field-Antifield Quantization 223

12.1 Introduction 223

12.2 Axiomatic field-antifield formalism 224

12.3 Constructive proof of the field-antifield formalism for a restricted class of theories 231

12.3.1 From the FV-phase-space action to the Hamiltonian master equation 232

12.3.2 Transition to configuration space 238

12.4 The Lagrangian master equation 247

12.5 The quantum master equation 253

12.5.1 An alternative derivation of the quantum master equation 256

12.5.2 Gauge invariant correlation functions 259

12.6 Anomalous gauge theories. The chiral Schwinger model 261

12.6.1 Quantum Master equation and the anomaly 265

A Local Symmetries and Singular Lagrangians 271

A.1 Local symmetry transformations 271

A.2 Bianchi identities and singular Lagrangians 274

B The BRST Charge of Rank One 278

C BRST Hamiltonian of Rank One 281

D The FV Principal Theorem 283

E BRST Quantization of SU (3) Yang-Mills Theory in α-gauges 287

Bibliography 291

Index 301