An Introduction to Non Perturbative Foundations of Quantum Field Theory 1st Edition by Franco Strocchi 9780191651342 0191651346

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ISBN 10: 0191651346
ISBN 13: 9780191651342
Author: Franco Strocchi

Quantum Field Theory (QFT) has proved to be the most useful strategy for the description of elementary particle interactions and as such is regarded as a fundamental part of modern theoretical physics. In most presentations, the emphasis is on the effectiveness of the theory in producing experimentally testable predictions, which at present essentially means Perturbative QFT. However, after more than fifty years of QFT, we still are in the embarrassing situation of not knowing a single non-trivial (even non-realistic) model of QFT in 3+1 dimensions, allowing a non-perturbative control. As a reaction to these consistency problems one may take the position that they are related to our ignorance of the physics of small distances and that QFT is only an effective theory, so that radically new ideas are needed for a consistent quantum theory of relativistic interactions (in 3+1 dimensions). The book starts by discussing the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series etc.). The general physical principles of positivity of the energy, Poincare’ covariance and locality provide a substitute for canonical quantization, qualify the non-perturbative foundation and lead to very relevant results, like the Spin-statistics theorem, TCP symmetry, a substitute for canonical quantization, non-canonical behaviour, the euclidean formulation at the basis of the functional integral approach, the non-perturbative definition of the S-matrix (LSZ, Haag-Ruelle-Buchholz theory). A characteristic feature of gauge field theories is Gauss’ law constraint. It is responsible for the conflict between locality of the charged fields and positivity, it yields the superselection of the (unbroken) gauge charges, provides a non-perturbative explanation of the Higgs mechanism in the local gauges, implies the infraparticle structure of the charged particles in QED and the breaking of the Lorentz group in the charged sectors. A non-perturbative proof of the Higgs mechanism is discussed in the Coulomb gauge: the vector bosons corresponding to the broken generators are massive and their two point function dominates the Goldstone spectrum, thus excluding the occurrence of massless Goldstone bosons. The solution of the U(1) problem in QCD, the theta vacuum structure and the inevitable breaking of the chiral symmetry in each theta sector are derived solely from the topology of the gauge group, without relying on the semiclassical instanton approximation.

An Introduction to Non Perturbative Foundations of Quantum Field Theory 1st Table of contents:

1. Relativistic quantum mechanics
2. Quantum mechanics and relativity
3. Relativistic Schrödinger wave mechanics
4. Relativistic Schrödinger equation
5. Klein–Gordon equation
6. Dirac equation
7. The general conflict between locality and energy positivity
8. Relativistic particle interactions and quantum mechanics
9. Problems of relativistic particle interactions
10. Field interactions and quantum mechanics
11. Free field equations and quantum mechanics
12. Particles as field quanta
13. Appendix: The Dirac equation
14. Appendix: Canonical field theory
15. Mathematical problems of the perturbative expansion
16. Dyson’s perturbative expansion
17. Dyson argument against convergence
18. φ4 model in zero dimensions
19. φ4 model in 0 + 1 dimensions
20. φ4 model in 1 + 1 and 2 + 1 dimensions
21. Haag theorem; non-Fock representations
22. Quantum field interacting with a classical source
23. Bloch–Nordsieck model; the infrared problem
24. Yukawa model; non-perturbative renormalization
25. Ultraviolet singularities and canonical quantization
26. Problems of the interaction picture
27. Appendix: Locality and scattering
28. Locality and asymptotic states
29. Scattering by a long-range potential
30. Adiabatic switching
31. Asymptotic condition
32. Wick theorem and Feynman diagrams
33. Compton and electron–electron scattering; electron–positron annihilation
34. Non-perturbative foundations of quantum field theory
35. Quantum mechanics and relativity
36. Properties of the vacuum correlation functions
37. Quantum mechanics from correlation functions
38. General properties
39. Spectral condition and forward tube analyticity
40. Lorentz covariance and extended analyticity
41. Locality and permuted extended analyticity
42. Local structure of QFT
43. Quantization from spectral condition
44. General non-perturbative results and examples
45. Free evolution implies canonical quantization
46. Spin–statistics theorem
47. PCT theorem
48. Appendix: PCT theorem for spinor fields
49. Haag theorem
50. Ultraviolet singularities and non-canonical behavior
51. Schwinger terms in current commutators
52. Axial current anomaly and π0 → 2γ decay
53. The derivative coupling model
54. Euclidean quantum field theory
55. The Schwinger functions
56. Euclidean invariance and symmetry
57. Reflection positivity
58. Cluster property
59. Laplace transform condition
60. From Euclidean to relativistic QFT
61. Examples
62. Functional integral representation
63. Non-perturbative S-matrix
64. LSZ asymptotic condition in QFT
65. Haag–Ruelle scattering theory (massive case)
66. One-body problem
67. Large time decay of smooth solutions
68. Refined cluster property
69. The asymptotic limit
70. The S-matrix and asymptotic completeness
71. Buchholz scattering theory (massless particles)
72. Huyghens’ principle and locality
73. One-body problem
74. Asymptotic limit
75. Remarks on the infrared problem
76. Quantization of gauge field theories
77. Physical counterpart of gauge symmetry
78. Gauss law and locality
79. Local gauge quantization of QED
80. Weak Gauss law
81. Subsidiary condition and gauge invariance
82. Indefinite metric and Hilbert–Krein structure
83. Charged states
84. Local gauge quantization of the Yang–Mills theory
85. Gauss law and charge superselection rule
86. Gauss charges in local gauges
87. Superselected charges and physical states
88. Electric charge, current, and photon mass
89. Gauss law and Higgs mechanism
90. Local gauges
91. Coulomb gauge; a theorem on the Higgs phenomenon
92. Delocalization and gap in Coulomb systems
93. Gauss law and infraparticles
94. Appendix: Quantization of the electromagnetic potential
95. Coulomb gauge
96. Feynman–Gupta–Bleuler quantization
97. Temporal gauge
98. Chiral symmetry breaking and vacuum structure in QCD
99. The U(1) problem
100. Topology and chiral symmetry breaking in QCD
101. Temporal gauge and Gauss law
102. Topology of the gauge group
103. Fermions and chiral symmetry
104. Solution of the U(1) problem
105. Topology and vacuum structure
106. Regular temporal gauge
107. A lesson from the Schwinger model

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