Quantum Field Theory (eBook)

CONTENTS 5
Preface 12
Constructive Use of Holographic Projections 19
1. Historical background and present motivations for holography 19
2. Lightfront holography, holography on null-surfaces and the origin of the area law 22
3. From holography to correspondence: the AdS/CFT correspondence and a controversy 32
4. Concluding remarks 40
Acknowledgements 41
References 41
Topos Theory and ‘Neo-Realist’ Quantum Theory 43
1. Introduction 43
2. A formal language for physics 49
3. The context category V(R) and the topos of presheaves SetV(R)op 51
4. Representing L(S) in the presheaf topos SetV(R)op 53
5. Truth objects and truth-values 56
6. Conclusion and outlook 63
Acknowledgements 63
References 64
A Survey on Mathematical Feynman Path Integrals: Construction, Asymptotics, Applications 66
1. Introduction 66
2. The mathematical realization of Feynman path integrals 69
3. Applications 73
Acknowledgements 79
References 79
A Comment on the Infra-Red Problem in the AdS/ CFT Correspondence 84
1. Introduction 84
2. Functional integrals on AdS 85
3. Two generating functionals 88
4. The infra-red problem and triviality 92
5. Conclusions and outlook 96
References 97
Some Steps Towards Noncommutative Mirror Symmetry on the Torus 99
1. Introduction 99
2. Elliptic curves 100
3. Noncommutative elliptic curves 101
4. Exotic deformations of the Fukaya category 104
5. Conclusion and outlook 107
Acknowledgements 108
References 108
Witten’s Volume Formula, Cohomological Pairings of Moduli Space of Flat Connections and Applications of Multiple Zeta Functions 111
1. Introduction 111
2. Background about moduli space 117
3. Volume of the moduli space of SU(2) flat connections 119
4. Volume of the moduli space of flat SU(3) connections 122
5. Cohomological pairings of the moduli space 124
References 130
Noncommutative Field Theories from a Deformation Point of View 133
1. Introduction 133
2. Noncommutative space-times 134
3. Matter fields and deformed vector bundles 137
4. Deformed principal bundles 141
5. The commutant and associated bundles 146
References 149
Renormalization of Gauge Fields using Hopf Algebras 152
1. Introduction 152
2. Preliminaries on perturbative quantum field theory 154
3. The Hopf algebra of Feynman graphs 158
4. The Hopf algebra of Green’s functions 162
Appendix A. Hopf algebras 167
References 168
Not so Non-Renormalizable Gravity 170
1. Introduction 170
2. The structure of Dyson–Schwinger Equations in QED4 171
3. Gravity 174
References 176
The Structure of Green Functions in Quantum Field Theory with a General State 178
1. Introduction 178
2. Expectation value of Heisenberg operators 180
3. QFT with a general state 181
4. Nonperturbative equations 183
5. Determination of the ground state 186
6. Conclusion 187
References 188
The Quantum Action Principle in the Framework of Causal Perturbation Theory 191
1. Introduction 191
2. The off-shell Master Ward Identity in classical field theory 193
3. Causal perturbation theory 196
4. Proper vertices 198
5. The Quantum Action Principle 200
6. Algebraic renormalization 208
References 209
Plane Wave Geometry and Quantum Physics 211
1. Introduction 211
2. A brief introduction to the geometry of plane wave metrics 212
3. The Lewis–Riesenfeld theory of the time-dependent quantum oscillator 221
4. A curious equivalence between two classes of Yang-Mills actions 225
References 229
Canonical Quantum Gravity and Effective Theory 231
1. Loop quantum gravity 231
2. Effective equations 234
3. A solvable model for cosmology 239
4. Effective quantum gravity 245
References 246
From Discrete Space-Time toMinkowski Space: Basic Mechanisms, Methods and Perspectives 249
1. Introduction 249
2. Fermion systems in discrete space-time 250
3. A variational principle 252
4. A mechanism of spontaneous symmetry breaking 254
5. Emergence of a discrete causal structure 257
6. A first connection to Minkowski space 259
7. A static and isotropic lattice model 263
8. Analysis of regularization tails 266
9. A variational principle for the masses of the Dirac seas 268
10. The continuum limit 270
11. Outlook and open problems 271
References 272
Towards a q-Deformed Quantum Field Theory 274
1. Introduction 274
2. q-Regularization 275
3. Basic ideas of the mathematical formalism 278
4. Applications to physics 287
5. Conclusion 294
References 294
Towards a q-Deformed Supersymmetric Field Theory 297
1. Introduction 297
2. Fundamental Algebraic Concepts 299
3. q-Deformed Superalgebras 302
4. q-Deformed Superspaces and Operator Representations 305
Appendix A. q-Analogs of Pauli matrices and spin matrices 310
References 312
L8-Algebra Connections and Applicationsto String- and Chern-Simons n-Transport 315
1. Introduction 315
2. The setting and plan 318
3. Statement of the main results 326
4. Differential graded-commutative algebra 330
5. L8-algebras and their String-like extensions 342
6. L8-algebra Cartan-Ehresmann connections 379
7. Higher string- and Chern-Simons n-bundles: the lifting problem 394
8. L8-algebra parallel transport 411
9. Physical applications: string-, fivebrane- and p-brane structures 426
Appendix A. Explicit formulas for 2-morphisms of L8-algebras 429
Acknowledgements 432
References 432
Index 437