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From Classical Field Theory to Perturbative Quantum Field Theory 2019 ed. [Kõva köide]

  • Formaat: Hardback, 536 pages, kõrgus x laius: 235x155 mm, kaal: 998 g, XXI, 536 p., 1 Hardback
  • Sari: Progress in Mathematical Physics 74
  • Ilmumisaeg: 27-Mar-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030047377
  • ISBN-13: 9783030047375
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From Classical Field Theory to Perturbative Quantum Field Theory 2019 ed.Suurem pilt
  • Formaat: Hardback, 536 pages, kõrgus x laius: 235x155 mm, kaal: 998 g, XXI, 536 p., 1 Hardback
  • Sari: Progress in Mathematical Physics 74
  • Ilmumisaeg: 27-Mar-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030047377
  • ISBN-13: 9783030047375
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783030047375.html
Märksõnad:

This book develops a novel approach to perturbative quantum field theory: starting with a perturbative formulation of classical field theory, quantization is achieved by means of deformation quantization of the underlying free theory and by applying the principle that as much of the classical structure as possible should be maintained.

The resulting formulation of perturbative quantum field theory is a version of the Epstein-Glaser renormalization that is conceptually clear, mathematically rigorous and pragmatically useful for physicists. The connection to traditional formulations of perturbative quantum field theory is also elaborated on, and the formalism is illustrated in a wealth of examples and exercises.

Arvustused

This book is a wonderful piece of scholarship, aimed at an audience of serious and correspondingly well-prepared mathematical physicists, or even mathematicians interested in this marvelous material and willing to play the long game. (Michael Berg, MAA Reviews, September 15, 2019)

Foreword xi
Klaus Fredenhagen
Preface xv
Introduction 1(2)
1 Classical Field Theory
1.1 Configuration space and the basic field
3(2)
1.2 The space of fields F
5(4)
1.3 Derivatives and support of functionals, local fields
9(7)
1.4 Balanced fields
16(2)
1.5 The field equation
18(3)
1.6 Retarded fields
21(4)
1.7 Perturbative expansion of retarded fields
25(2)
1.8 The Poisson algebra of the free theory
27(4)
1.9 An explicit formula for the classical retarded product
31(7)
1.10 Basic properties of the retarded product
38(8)
2 Deformation Quantization of the Free Theory
2.1 Star products of fields
46(3)
2.2 Solutions for the two-point function Hm
49(4)
2.3 Equivalence of the several star products
53(4)
2.4 Existence and properties of the star product
57(3)
2.5 States
60(3)
2.6 Connection to the algebra of Wick polynomials
63(9)
3 Perturbative Quantum Field Theory
3.1 Axioms for the retarded product
72(35)
3.1.1 Basic axioms
73(6)
3.1.2 A worked example
79(1)
3.1.3 Discussion of the GLZ relation
80(3)
3.1.4 Further axioms: Renormalization conditions
83(8)
3.1.5 The scaling and mass expansion
91(12)
3.1.6 The classical limit
103(1)
3.1.7 (Feynman) diagrams
104(3)
3.2 Construction of the retarded product
107(52)
3.2.1 Inductive step, off the thin diagonal Δn
107(7)
3.2.2 The extension to the thin diagonal Δn
114(15)
3.2.3 Maintaining scaling and mass expansion in the extension
129(1)
3.2.4 Completing the inductive step
130(6)
3.2.5 Scaling degree axiom versus Sm-expansion axiom
136(3)
3.2.6 The general solution of the axioms
139(6)
3.2.7 Maintaining symmetries in the extension of distributions
145(7)
3.2.8 Explicit computation of an interacting field -- renormalization of two-point functions using the Kallen--Lehmann representation
152(7)
3.3 The time-ordered product
159(21)
3.3.1 Heuristic explanation of the physical relevance of the time-ordered product
159(2)
3.3.2 Axioms for the time-ordered product and its inductive construction
161(10)
3.3.3 Connection between the T- and the R-product
171(9)
3.4 The time-ordered (or retarded) product for on-shell fields
180(13)
3.4.1 Definition of the on-shell T-product
180(5)
3.4.2 Properties of the on-shell T-product
185(8)
3.5 Techniques to renormalize in practice
193(26)
3.5.1 Differential renormalization
193(9)
3.5.2 Analytic regularization
202(17)
3.6 The Stuckelberg--Petermann renormalization group
219(23)
3.6.1 Definition of the Stuckelberg--Petermann RG
220(5)
3.6.2 The Main Theorem of perturbative renormalization
225(9)
3.6.3 The Gell-Mann--Low cocycle
234(8)
3.7 The algebraic adiabatic limit
242(6)
3.8 The renormalization group in the algebraic adiabatic limit
248(23)
3.8.1 Renormalization of the interaction
248(7)
3.8.2 Wave function, mass and coupling constant renormalization
255(10)
3.8.3 Field renormalization
265(6)
3.9 A version of Wilson's renormalization group and of the flow equation
271(29)
3.9.1 Outline of the procedure
271(3)
3.9.2 The regularized S-matrix
274(12)
3.9.3 Effective potential and flow equation
286(6)
3.9.4 A version of Wilson's renormalization "group"
292(1)
3.9.5 Comparison with the functional integral approach
293(7)
4 Symmetries -- the Master Ward Identity
4.1 Derivation of the Master Ward Identity in classical field theory
300(1)
4.2 The Master Ward Identity as a universal renormalization condition
301(13)
4.2.1 Formulation of the MWI
301(1)
4.2.2 Verification that the MWI is a renormalization condition
302(2)
4.2.3 A few applications of the MWI
304(10)
4.3 Anomalous Master Ward Identity (Quantum Action Principle)
314(9)
4.4 Reduction of the MWI to the "quantum part"
323(13)
4.4.1 Proper vertices
323(10)
4.4.2 The Master Ward Identity in terms of proper vertices
333(3)
4.5 Removing violations of the Master Ward Identity
336(14)
4.5.1 Proceeding analogously to algebraic renormalization
336(9)
4.5.2 Proof of the relevant MWI for the scalar O(N)-model
345(5)
5 Quantum Electrodynamics
5.1 Deformation quantization for fermionic and gauge fields
350(53)
5.1.1 Dirac spinors
350(23)
5.1.2 Faddeev--Popov ghosts: A pair of anticommuting scalar fields
373(2)
5.1.3 The photon field
375(3)
5.1.4 Definition of the retarded and time-ordered product in QED
378(4)
5.1.5 Charge conjugation invariance
382(4)
5.1.6 The PCT-Theorem
386(10)
5.1.7 Reducing the computation of R- and T-products to scalar field renormalization
396(7)
5.2 The relevant Master Ward Identity for Quantum Electrodynamics
403(13)
5.2.1 Formulation of the QED Master Ward Identity
403(4)
5.2.2 Proof of the QED-MWI
407(6)
5.2.3 Conservation of the interacting current and the corresponding charge
413(3)
5.3 The local algebras of interacting fields
416(3)
5.4 Connection of observable algebras and field algebras
419(17)
5.4.1 Local construction of observables in gauge theories
419(5)
5.4.2 Construction of physical states on the algebra of observables
424(3)
5.4.3 Stability under deformations
427(9)
5.5 Verification of the assumptions for QED
436(24)
5.5.1 The free theory
436(14)
5.5.2 The interacting theory: The interacting Kugo--Ojima charge
450(10)
5.6 Reasons in favour of a LOCAL construction
460(3)
Appendix
A.1 Some notations and a few mathematical preliminaries
463(6)
A.2 Propagators: Conventions and properties
469(4)
A.3 A short introduction to wave front sets
473(5)
A.4 Perturbative QFT based on quantization with a Hadamard function
478(6)
A.5 The Fock space
484(11)
A.6 Weak adiabatic limit: Wightman- and Green functions
495(10)
A.7 Remarks about the connection to traditional approaches
505(8)
Glossary of Abbreviations and Symbols 513(6)
Bibliography 519(12)
Index 531