Rigorous Quantum Field Theory (eBook)

Contents 6
1 Introduction 8
References 16
2 Local Counterterms on the Noncommutative Minkowski Space 18
2.1 Free fields and perturbation theory 19
2.2 Local counterterms 26
2.3 Quasiplanar Wick products 29
2.4 Outlook 32
References 32
3 Massless Scalar Field in a Two-dimensional de Sitter Universe 34
3.1 de Sitter geometry and the massive scalar field 34
3.2 Massless limit 36
3.3 Invariant Krein space 37
3.4 Equation of motion and gauge invariance 41
3.5 Conclusions 44
References 45
4 Locally Covariant Quantum Field Theories 46
4.1 Locality and general covariance 46
4.2 Quantum Field Theory as a Functor 47
4.3 Beyond simple functoriality: Equivalence, Dynamics, Fields, Scattering, and more. 49
4.4 Conclusions and Outlook 53
References 53
5 Asymptotic Abelianness and Braided Tensor C* - Categories 56
5.1 Introduction 56
5.2 Asymptotically Abelian Intertwiners 58
5.3 The Emergence of Braiding 64
5.4 Algebraic Quantum Field Theory 67
Acknowledgements 70
References 70
6 Yang–Mills and Some Related Algebras 72
6.1 Introduction 72
6.2 Homogeneous algebras 73
6.3 The Yang–Mills algebra 76
6.4 The super Yang–Mills algebra 78
6.5 The super self-duality algebra 80
6.6 Deformations 83
References 85
7 Remarks on the Anti-de Sitter Space-Time 86
7.1 Introduction 86
7.2 Some Lorentzian geometry in AdS 89
7.3 Real Scalar Field on 91
7.4 The n - point Tuboids 93
7.5 Spectral Condition for AdS 94
7.6 Two-Point Functions, Generalized Free Fields 95
7.7 n- Point Permuted Tuboids 96
7.8 Bisognano-Wichmann-KMS Property, CTP 97
7.9 Wick Rotations for Xd 98
Appendix. q-Sheeted AdS Covers and Exotic Locality 98
References 100
8 Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory 102
8.1 Introduction 102
8.2 Quantum Energy Inequalities 103
8.3 Stability at Three Scales 105
8.4 Connections with Nuclearity 112
8.5 Conclusion 115
References 116
9 Action Ward Identity and the Stückelberg–Petermann Renormalization Group 120
9.1 Introduction 120
9.2 Basic properties required for interacting fields 121
9.3 Construction of solutions 123
9.4 Renormalization group 125
9.5 Local nets and local fields 126
References 129
10 On the Relativistic KMS Condition for the Model 132
10.1 Introduction 132
10.2 The spatially-cutoff model at positive temperature 133
10.3 Euclidean approach 137
10.4 The relativistic KMS condition 140
10.5 Outlook 146
References 146
11 The Analyticity Program in Axiomatic Quantum Field Theory 148
11.1 Introduction 148
11.2 The axiomatic framework 149
11.3 Causality and Analyticity: mathematical results 153
11.4 The linear axiomatic program 156
11.5 Causality and local analyticity of chronological functions: physical discussion 160
11.6 The nonlinear program: some results 162
11.7 The Analyticity Program in Constructive QFT 165
12 Renormalization Theory Based on Flow Equations 168
12.1 Introduction 168
12.2 Renormalization of .4 Theory 169
12.3 Relativistic Theory 175
12.4 A short look at further results 178
References 179
13 Towards the Construction of Quantum Field Theories from a Factorizing S- Matrix 182
13.1 Introduction 182
13.2 Wedge-local fields 184
13.3 Existence of local observables 187
13.4 Modular Compactness for Wedge Algebras 190
13.5 Summary and Outlook 200
Appendix 200
References 202
14 String-Localized Covariant Quantum Fields 206
14.1 Introduction 206
14.2 Wigner Particles 208
14.3 Modular Localization Structure for 209
14.4 String-Localized Covariant Wave Functions 211
14.5 Summary and Outlook 214
A Extension of the Representations to P+ 215
B Intertwiners and Localization Structure for the Principal Series Representations 216
References 219
15 Quantum Anosov Systems 220
15.1 Introduction 220
15.2 The classical system 221
15.3 Spectral properties and clustering properties 222
15.4 Verification of the Anosov structure in crossed product constructions 223
15.5 The Type III Case 225
15.6 Application to Quantum field Theory 228
References 229
16 DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction 232
16.1 Introduction 232
16.2 DFR Quantum Space-time, and All That 233
16.3 Nonlocal Dyson Diagrams 238
16.4 Conclusions 241
Appendix. Twisted Products 241
References 243
17 On Local Boundary CFT and Non- Local CFT on the Boundary 246
17.1 Introduction 246
17.2 Algebraic boundary conformal QFT 247
17.3 The charge structure of BCFT fields 252
17.4 Nimreps and non-vacuum BCFT 253
17.5 Appendix: Modular Theory in QFT and in BCFT 256
References 258
18 Algebraic Holography in Asymptotically Simple, Asymptotically AdS Space- times 260
18.1 Introduction 260
18.2 Doing away with coordinates in Rehren duality 262
18.3 Properties of the Rehren bijection in AAdS space-times 267
18.4 Perspectives and open problems 275
Acknowledgements 276
References 276
19 Non-Commutative Renormalization 278
19.1 Introduction 278
19.2 The UV-IR Problem 280
19.3 The Covariant Theory 281
19.4 Smooth slices 285
19.5 Towards Non-commutative Constructive Field Theory 286
References 288
20 New Constructions in Local Quantum Physics 290
20.1 How modular theory entered particle physics 290
20.2 Modular localization and the bootstrap-formfactor program 295
20.3 Constructive aspects of lightfront holography 300
References 306
21 Physical Fields in QED 308
21.1 Introduction 308
21.2 Formal Considerations 309
21.3 Problems, and a Solution 310
Appendix: Proving Equation (21.17) 314
References 315
22 Complex Angular Momentum Analysis and Diagonalization of the Bethe – Salpeter Structure in Axiomatic Quantum Field Theory 318
22.1 Introduction: Phenomenological Motivation 318
22.2 Harmonic Analysis on Complex Hyperboloids 320
22.3 Complex Angular Momentum in General Quantum Field Theory 326
References 332

1 Introduction (P. 1)

After the great scienti.c revolutions of Special Relativity and of Quantum Mechanics in the first half of the twentieth century, Relativistic Quantum Field Theory (QFT) was introduced to provide a synthesis of these two new paradigms (in Kuhn’s words). As such, QFT is one of the major advances for theoretical physics in the second half of the last century.

The main reason for emphasizing the importance of QFT lies in the impressive effort that it represents towards a unified understanding of the structure of matter at subatomic scales, as it emerges in the collision phenomena of particle physics in the range of energies presently explored in particle accelerators. But another reason is the extraordinary mathematical wealth of this theoretical framework, which makes it a fascinating domain of research for mathematical physics and may also stimulate the interest of mathematicians.

Foreseen by Dirac in 1933 and finaly established discovered by Feynman around 1949 for the treatment of quantum electrodynamics, the so-called "path-integral formalism" of quantum field theory is also now considered as a powerful tool for providing perturbative methods of computation in many problems of theoretical physics, namely in statistical mechanics and in string theory. However, the use of relativistic quantum fields as a basic concept of mathematical physics underlying all the phenomena of particle physics in a very large range of energies represents a much more ambitious program.

This program was indeed stimulated by the success of the quantum electrodynamics (QED) formalism for computing the electron-photon, electronelectron and electron-positron scattering amplitudes. Even today, it is by no means understood why the perturbative expansion of QED in powers of the coupling parameter, that is the electric charge of the electron, provides such a spectacular agreement with experimental data, although in practice it is reduced to the computation of the very first terms of the series (the only ones which are computable).

This is even more surprising since we now know that the series cannot be convergent, so that, paraphrasing Wigner’s words, one may wonder about the "unreasonable effectiveness" of perturbation theory in four-dimensional QED. In the 1950s, however, the success of these computations was credited to the smallness of the coupling parameter of QED, and this situation stimulated research for other methods of investigation of QFT, which could apply to quantum field models with large coupling: it was in fact the very concept of a quantum field which appeared as the most powerful and promising one for explaining the phenomena of strong nuclear interaction of particle physics.

During all that period the study of phenomena of weak nuclear interaction also benefitted, like QED, from the success of a perturbative QFT formalism reduced to very few terms. The demand for a more general nonperturbative treatment of QFT motivated an important community of mathematical physicists to work out a model-independent axiomatic approach to relativistic quantum field theory [2,8,9,11,14].

Their first task was to provide a mathematically meaningful concept of a relativistic quantum field in terms of "operator-valued distribution in the Hilbert space of quantum physical states", in such a way that the notion of ingoing and outgoing particle states could be introduced in terms of appropriate asymptotic forms of the field operators.