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E-raamat: Non-perturbative Methods In Two Dimensional Quantum Field Theory

(Univ Heidelberg, Germany), (Univ Estadual Paulista, Brazil), (Univ De Sao Paulo, Brazil)
  • Formaat: 748 pages
  • Ilmumisaeg: 12-Aug-1991
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814506519
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Non-perturbative Methods In Two Dimensional Quantum Field TheorySuurem pilt
  • Formaat: 748 pages
  • Ilmumisaeg: 12-Aug-1991
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814506519
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This book is a survey of methods used in the study of two-dimensional models in quantum field theory as well as applications of these theories in physics. It covers the subject since the first model, studied in the fifties, up to modern developments in string theories, and includes exact solutions, non-perturbative methods of study, and nonlinear sigma models.

Arvustused

"The authors have made a great effort to present in a careful and systematic way such a complete treatise for the benefit of non-specialists. This book is thus of great value to students and research physicists interested in two-dimensional quantum field theory. Numerous (15) useful appendices help the reader to understand and rederive the results at will, making the book self-contained." Jean-Pierre Ader Mathematical Reviews

Foreword vii
Introduction
1(642)
Free Fields
Introduction
7(1)
Bosonic Free Fields
7(5)
Fermionic Free Fields
12(3)
Bosonization of Fermions
15(5)
The RS-Model
20(3)
Conclusions
23(1)
The Thirring Model
Introduction
24(1)
The Massless Thirring Model
25(4)
The Massive Thirring Model
29(9)
Equivalence with sine-Gordon equation
29(2)
Classical conservation laws.
31(2)
Quantum conservation laws
33(5)
Bosonization Revisited
38(3)
Fermions in terms of bosons
39(2)
The Soliton as a Disorder Parameter
41(7)
Conclusion
48(1)
Functional Determinants
Definition and General Remarks
49(4)
Determinants and the Generalized Zeta-Function
53(6)
Elliptic operators
53(2)
The zeta-function
55(4)
One Point Compactification
59(9)
Stereographic projection of Dirac operator
59(5)
The associated Dirac operator
64(2)
Vortex on the stereographic sphere
66(2)
Calculation of Seeley Coefficients
68(4)
Methods for Computing Determinants
72(16)
ξ-function regularization
72(3)
Proper-time regularization
75(3)
Fujikawa method
78(4)
Determinant from current-anomaly
82(5)
Pauli-representation of determinant
87(1)
Ambiguities in the Calculation of the Determinant
88(5)
Ambiguities in the regularization
88(2)
Dependence on R
90(3)
Atiyah-Singer Index Theorem
93(2)
Conclusion
95(2)
Fermionic Models with Self-interaction
Introduction
97(1)
The O(N) Invariant Gross-Neveu Model
97(16)
Classical conservation laws
98(2)
Effective potential and β-function in a 1/N expansion
100(4)
1/N expansion: Feynman rules
104(2)
Leading order S-matrix elements
106(3)
Quantization of non-local charge
109(4)
Chiral Gross-Neveu Model
113(12)
Cancellation of infrared singularities
114(2)
The 1/N expansion
116(2)
Operator formulation
118(6)
Quantization of non-local charge
124(1)
Conclusions and Physical Interpretation
125(1)
Non-linear Sigma Models- Classical Aspects
Introduction
126(12)
Historical development
126(1)
Sigma models and current algebra
127(2)
Two-dimensional sigma models: preliminaries
129(9)
Purely Bosonic Non-linear Sigma Models
138(21)
Formal development
138(8)
Dual symmetry and higher conservation laws
146(10)
An explicit example: the Grassmannians
156(3)
Non-linear Sigma Models with Fermions
159(17)
Definition and properties
159(5)
Dual symmetry and higher conservation laws
164(7)
Construction of an explicit example
171(5)
Analogies with Four Dimensional Gauge Theories
176(5)
Concluding Remarks
181(5)
Non-linear Sigma Models - Quantum Aspects
Introduction
186(1)
Grassmannian Bosonic Models
187(12)
1/N expansion
187(6)
Renormalization
193(3)
Infrared divergencies
196(2)
Physical interpretation of the results
198(1)
Grassmannian Models Interacting with Fermions
199(13)
1/N expansion and Feynman rules
199(6)
Physical interpretation of the results
205(7)
Quantization of Higher Conservation Laws
212(10)
Purely bosonic sigma models and anomalies
212(6)
Fermionic interaction and anomaly cancellation
218(4)
Perturbative Renormalization and the Background Field Method
222(9)
Background field method applied to the sigma model
222(6)
Parallelizable manifolds, and application to string theory
228(3)
Anomalous Non-Linear Sigma Models in Four Dimensions
231(1)
Conclusion
232(2)
Exact S-matrices of Two Dimensional Models
Introduction
234(9)
Consequences of higher conservation laws
234(2)
Factorizable S-matrix
236(3)
Fusion rules
239(3)
Bound state scattering
242(1)
Classification of Exact S-matrices from the Local Conservation Laws
243(10)
SU(N) invariant S-matrix
243(1)
S-matrix of the sine-Gordon and massive Thirring model
244(7)
Exact S-matrix for O(N) symmetry
251(1)
The ZN invariant S-matrix
251(2)
Quantum Non-Local Charges, and Exact S-Matrices
253(15)
S-matrices of purely fermionic models
254(4)
S-matrices of non-linear sigma models
258(10)
Further Developments and Conclusion
268(3)
The Wess-Zumino-Witten Theory
Introduction
271(3)
Existence of a Critical Point
274(3)
Properties at the Critical Point
277(8)
The Kac Moody algebra
277(4)
The WZW fields in terms of fermions
281(1)
The Sugawara form of the energy momentum tensor
281(2)
The non-abelian bosonization in the operator language
283(2)
Properties off the Critical Point
285(6)
Integrability of the Wess-Zumino-Witten action
285(1)
On the solution off the critical point
286(2)
Supersymmetric WZW model
288(3)
Conclusion
291(1)
Quantum Electrodynamics: Operator Approach
Introduction
292(2)
The Massless Schwinger Model
294(28)
Quantum solution
294(3)
The Maxwell current
297(3)
Chiral densities
300(1)
Vacuum structure
301(5)
Gauge transformations
306(2)
Correlation functions and violation of clustering
308(2)
Absence of charged states (screening)
310(2)
The quark-antiquark potential
312(2)
Adding flavor
314(4)
Fractional winding number and the U(1) problem
318(4)
The Massive Schwinger Model
322(30)
Equivalent bosonic formulation
322(3)
The quantum Dirac equation
325(3)
Vacuum structure and all that
328(1)
Screening versus confinement
329(9)
Adding flavour
338(7)
Lorentz transformation properties
345(3)
The MSM as the limit of a massive vector theory
348(4)
Conclusion
352(1)
Quantum Chromodynamics
Introduction
353(4)
The External Field Current
357(8)
Tree graph expansion
357(3)
Loop expansion
360(5)
QCD2 Determinant
365(8)
Heat-kernel method
365(4)
Polyakov-Wiegman method
369(2)
A further invariance of the effective action
371(2)
Bosonization of the QCD2 Fermionic Action
373(3)
Recovering the U(1) case
374(1)
QCD2 bosonized currents
375(1)
Fermion Greens Function
376(2)
Specializing to the U(1) case
377(1)
Conclusion
378(3)
Quantum Electrodynamics: Functional Approach
Introduction
381(1)
Equivalent Bosonic Action
382(1)
Gauge Invariant Correlation Functions
383(2)
The external field current and chiral densities
384(1)
Vacuum Structure
385(5)
Chirality of the vacuum
385(5)
Why Study Gauge Invariant Correlation Functions?
390(1)
Screening versus Confinement
391(3)
Quasi Periodic Boundary Conditions and the θ Vacuum
394(3)
Functional Representation of Tunneling Amplitudes
397(4)
Interpretation of the Result
401(7)
Zero modes
403(2)
Calculations of the det i D from the anomaly equation
405(3)
Eigenvalue Spectrum of the Associated Dirac Operator
408(3)
Zero Modes of the Dirac Operator as Boundary-Valued Problem
411(8)
The U(1) Problem Revisited
419(6)
Conclusion
425(2)
Non-Abelian Chiral Gauge Theories
Introduction
427(5)
Anomalies and Cocycles
432(12)
Consistent anomaly
432(5)
More about cocycles
437(2)
Gauss anomaly
439(2)
Relation between consistent and covariant anomaly
441(3)
Isomorphic Representations of Chiral QCD2
444(4)
External Field Ward Identities
448(1)
Ward-Takahashi Identities
449(3)
The Effective Bosonic Action in the GNI Formulation
452(3)
Construction of the One-Cocycle from the Anomaly
455(1)
Bosonic Action in the GNI and GI Formulation
456(4)
Symmetries of the Model
460(2)
Relation of Source Currents in GNI and GI Formulations
462(2)
Poisson Algebra of the Currents
464(4)
Hamiltonian Quantization
468(10)
Fermionization of α1[ A,g-1]
478(1)
BRS Quantization of GI Formulation
479(8)
Chiral QCD2 in terms of Chiral Bosons
487(6)
Constraint Structure from Fermionic Hamiltonian
493(8)
Conclusion
501(2)
Chiral Quantum Electrodynamics
Introduction
503(1)
The JR Model
504(3)
Quantization in GNI Formulation
507(8)
Hamiltonian and constraints
507(2)
Commutation relations
509(2)
Current-potential and bosonic representation of the fermion field
511(2)
Energy-momentum tensor
513(1)
Aμ two-point function
514(1)
Fermionic two-point function
515(1)
Quantization in the GI Formulation
515(10)
Hamiltonian and constraints
515(2)
Implementation of gauge conditions
517(2)
Isomorphism between GI and GNI formulation: phase space view
519(3)
Alternative approach to quantization
522(3)
Path-Integral Formulation
525(7)
Perturbative Analysis in the Fermionic Formulation
532(9)
Perturbative analysis in the GNI formulation
532(6)
Perturbative analysis in the GI formulation
538(3)
Anomalous Poisson Brackets Revisited
541(6)
Operator view of anomalous Poisson brackets
541(2)
Bjorken-Johnson-Low view of anomalous Poisson brackets
543(1)
Reconstruction of commutators of the GNI formulation
544(3)
Chiral QED2 in terms of Chiral Bosons
547(3)
Conclusion
550(2)
Conformally Invariant Field Theory in Two Dimensions
Introduction
552(1)
Conformal Transformations in Field Theory
553(18)
The conformal algebra
553(5)
The BPZ construction
558(13)
Operator Realization of the Conformal Algebra
571(6)
Non-Abelian Conformal Theory
577(11)
WZNW theory revisited
577(1)
Kac Moody algebra
577(3)
Ward identities and equations of motion
580(2)
Correlation functions
582(6)
The Non-Abelian Thirring Model at the Critical Point, and the WZW Theory
588(6)
Definition of the conformally invariant theory
588(2)
The solution at the conformally invariant fixed point
590(4)
More on Kac Moody Algebras
594(4)
Further Applications to String Theories
598(3)
Superconformal Symmetry
601(4)
Conformally Invariant Two-Dimensional Gravity
605(22)
The geneal theory of gravity in two dimensions
605(1)
Induced gravity; contribution from the matter fields to the determinant
606(9)
Canonical quantization and SL(2, R) symmetry
615(7)
Operator product expansions and quantum solution
622(2)
Interaction of matter fields with gravity in two dimensions
624(3)
Two-Dimensional Supergravity
627(5)
General theory of supergravity in two dimensions
627(2)
OPE and quantum solution
629(2)
Interaction of matter fields with supergravity
631(1)
Non-Perturbative 2D Gravity, Surfaces, and Critical Behavior
632(8)
Conclusions
640(3)
Final Remarks
643(4)
Appendix A Notation (Minkowski Space) 647(6)
Appendix B Notation (Euclidean Space) 653(3)
Appendix C Further Conventions 656(4)
Appendix D Functional Bosonization of the Massive Thirring Model 660(3)
Appendix E Bosonization of the Kinetic Term 663(2)
Appendix F Classical Integrability in the Massive Thirring Model 665(2)
Appendix G Quantum Non-Local Charge: Action on Asymptotic States 667(3)
Appendix H Non-Linear Sigma Models 670(3)
Appendix I Conservation Laws in NLS Models 673(1)
Appendix J S-Matrices 674(3)
Appendix K Complete S-Matrix of the Gross-Neveu Model 677(3)
Appendix L Poisson Brackets and Commutators 680(2)
Appendix M Chiral Bosons 682(6)
Appendix N Teichmuller Parameters 688(2)
Appendix O Dirac Quantization 690(12)
References 702(23)
Index 725
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