Gauging What's Real: The Conceptual Foundations of Contemporary Gauge Theories Book

Preface ix

Acknowledgements xiii

Introduction xv

1 What is a gauge theory? 1

1.1 Classical electromagnetism: a paradigm gauge theory 3

1.2 A fiber bundle formulation 7

1.2.1 Electromagnetic interactions of quantum particles 14

1.2.2 Electromagnetic interactions of matter fields 18

2 The Aharonov-Bohm effect 21

2.1 Fiber bundles 26

2.2 A gauge-invariant, local explanation? 31

2.3 Geometry and topology in the Aharonov-Bohm effect 40

2.4 Locality in the Aharonov-Bohm effect 44

2.5 Lessons for classical electromagnetism 54

3 Classical gauge theories 58

3.1 Non-Abelian Yang-Mills theories 58

3.1.1 The fiber bundle formulation 64

3.1.2 Loops, groups, and hoops 70

3.1.3 Topological issues 74

3.2 A fiber bundle formulation of general relativity 77

3.2.1 A gravitational analog to the Aharonov-Bohm effect 78

4 Interpreting classical gauge theories 82

4.1 The no gauge potential properties view 83

4.2 The localized gauge potential properties view 85

4.2.1 Problems defining theoretical terms 91

4.2.2 Leeds's view 99

4.2.3 Maudlin's interpretation 102

4.3 The non-localized gauge potential properties view 104

4.4 A holonomy interpretation 111

4.4.1 Epistemological considerations 112

4.4.2 Objections considered 119

4.4.3 Semantic considerations 122

4.5 Metaphysical implications: non-separability and holism 123

5 Quantized Yang-Mills gauge theories 129

5.1 How to quantize a classical field 131

5.2 Coulomb gauge quantization 133

5.3 Lorenz gauge quantization 135

5.4 Classical electromagnetism as a constrained Hamiltonian system 136

5.5 The free Maxwell field as a Hamiltonian system 139

5.6 Path-integral quantization141

5.7 Canonical quantization of non-Abelian fields 143

5.8 Path-integral quantization of non-Abelian fields 145

5.9 Interacting fields in the Lagrangian formulation 146

6 The empirical import of gauge symmetry 149

6.1 Two kinds of symmetry 150

6.2 Observing gauge symmetry? 155

6.3 The gauge argument 159

6.4 Ghost fields 167

6.5 Spontaneous symmetry-breaking 169

6.6 The θ-vacuum 175

6.7 Anomalies 182

7 Loop representations 184

7.1 The significance of loop representations 185

7.2 Loop representations of the free Maxwell field 186

7.3 Loop representations of other free Yang-Mills fields 192

7.4 Interacting fields in loop representations 195

7.5 The θ-vacuum in a loop representation 197

7.6 Conclusion 198

8 Interpreting quantized Yang-Mills gauge theories 200

8.1 Auyang's event ontology 200

8.2 Problems of interpreting a quantum field theory 203

8.2.1 Particle interpretations 205

8.2.2 Bohmian interpretations 209

8.2.3 Copenhagen interpretations 212

8.2.4 Everettian interpretations 215

8.2.5 Modal interpretations 218

9 Conclusions 220

A Electromagnetism and its generalizations 229

B Fiber bundles 233

C The constrained Hamiltonian formalism 248

D Alternative quantum representations 257

E Algebraic quantum field theory 265

F Interpretations of quantum mechanics 272

F.1 The Copenhagen interpretation 272

F.2 Bohmian mechanics 274

F.3 Everettian interpretations 276

F.4 Modal interpretations 278

Bibliography 280

Index 287