Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (2nd Edition)

By: Elcio Abdalla (Editor), Klaus D. Rothe (Editor), Maria Cristina Batoni Abdalla (Editor)

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Hardcover | 3 August 2001 | Edition Number 2

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836 Pages

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25.4 x 17.78 x 5.08

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The second edition of "Non-Perturbative Methods in Two-Dimensional Quantum Field Theory" is an extensively revised version, involving major changes and additions. Although much of the material is special to two dimensions, the techniques used should prove helpful also in the development of techniques applicable in higher dimensions. In particular, the last three chapters of the book should be of direct interest to researchers wanting to work in the field of conformal field theory and strings. This book is intended for students working for their PhD degree and post-doctoral researchers wishing to acquaint themselves with the non-perturbative aspects of quantum field theory.

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Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Classical Mathematics Wave Mechanics including Vibration & Acoustics Booktopia Publisher Services World Scientific Publishing Particle & High-Energy Physics

Introduction	p. 17
Free Fields	p. 25
Introduction	p. 25
Bosonic Free Fields	p. 25
Fermionic Free Fields	p. 29
Bosonization of Massless Fermions	p. 31
The RS-Model	p. 35
Conclusions	p. 38
The Thirring model	p. 41
Introduction	p. 41
The Massless Thirring Model	p. 42
The Massive Thirring Model	p. 45
Equivalence with sine-Gordon equation	p. 45
Classical conservation laws	p. 47
Quantum conservation laws	p. 48
Bosonization Revisited	p. 53
Fermions in terms of bosons	p. 53
The Soliton as a Disorder Parameter	p. 55
Conclusion	p. 61
Determinants and Heat Kernels	p. 65
Introduction	p. 65
Functional Determinant, one-loop diagram	p. 66
Determinants and the Generalized Zeta-Function	p. 69
One Point Compactification	p. 75
The associated Dirac operator	p. 78
Calculating Seeley Coefficients	p. 81
The perturbative approach	p. 81
The Schwinger-DeWitt method	p. 83
The Fujikawa method	p. 86
Computing Functional Determinants	p. 88
[zeta]-function regularization	p. 88
Proper-time regularization	p. 90
The Fujikawa point of view	p. 91
A Theorem on a one parameter family of factorizable operators	p. 95
The QCD[subscript 2] functional determinant	p. 98
Zero-modes	p. 101
Axial anomaly equation in the presence of zero-modes	p. 101
Atiyah-Singer Index Theorem	p. 104
Ambiguities in Functional Determinants	p. 107
Ambiguities in the regularization	p. 107
Dependence on the scale parameter	p. 108
Mass expansion in proper-time regularization	p. 111
The Finite Temperature Heat Kernel	p. 115
Scalar field in a static background potential	p. 117
Scalar field in a static background gauge potential	p. 119
Conclusion	p. 123
Self-Interacting fermionic models	p. 127
Introduction	p. 127
The O(N) Invariant Gross--Neveu Model	p. 127
Classical conservation laws	p. 128
Effective potential and [beta]-function in a 1/N expansion	p. 129
The 1/N Expansion: Feynman rules	p. 133
Leading order S-matrix elements	p. 135
Quantization of the non-local charge	p. 138
Chiral Gross--Neveu Model	p. 141
Cancellation of infrared singularities	p. 142
The 1/N expansion	p. 144
Operator formulation	p. 146
Quantization of non-local charge	p. 151
Conclusions and Physical Interpretation	p. 152
Non-linear [sigma] Models: Classical Aspects	p. 155
Historical development	p. 155
Sigma models and current algebra	p. 156
Two-dimensional [sigma] models: preliminaries	p. 158
Purely Bosonic Non-linear [sigma] Models	p. 166
Formal developments	p. 166
Dual symmetry and higher conservation laws	p. 172
An explicit example: the Grassmannians	p. 182
Non-linear [sigma] Models with Fermions	p. 184
Definition and properties	p. 184
Dual symmetry and higher conservation laws	p. 188
Construction of an explicit example	p. 195
Analogies with 4D Gauge Theories	p. 200
Concluding Remarks	p. 204
Non-linear [sigma] Models - Quantum Aspects	p. 211
Introduction	p. 211
Grassmannian Bosonic Models	p. 212
1/N expansion	p. 212
Renormalization	p. 218
Infrared divergencies	p. 219
Physical interpretation of the results	p. 221
Grassmannian Models and Fermions	p. 222
1/N expansion and Feynman rules	p. 222
Physical interpretation of the results	p. 227
Quantization of Higher Conservation Laws	p. 233
Purely bosonic sigma models and anomalies	p. 233
Fermionic interaction and anomaly cancellation	p. 239
Algebra of non-local charges	p. 242
Bosonic O(N)-symmetric sigma models	p. 242
Non-local charges in the WZNW model	p. 254
Perturbative Renormalization	p. 257
Background Field Method	p. 257
Parallelizable manifolds; applications to string theory	p. 263
Anomalous Non-Linear [sigma] Models in four dimensions	p. 265
Conclusion	p. 266
Exact S-matrices of 2D Models	p. 273
Introduction	p. 273
Consequences of higher conservation laws	p. 273
Factorizable S-matrix	p. 274
Fusion rules	p. 278
Bound state scattering	p. 280
S-matrices and Conservation Laws	p. 280
SU(N) invariant S-matrices	p. 280
Sine-Gordon and massive Thirring models	p. 282
Exact S-matrix for O(N) symmetry	p. 287
The Z[subscript N] invariant S-matrix	p. 288
Quantum Non-Local Charges and S-Matrices	p. 289
S-matrices of purely fermionic models	p. 289
S-matrices of non-linear sigma models	p. 293
Boundary S-matrices	p. 303
Further Developments	p. 307
Conclusion	p. 308
The Wess--Zumino--Witten Theory	p. 313
Introduction	p. 313
Existence of a Critical Point	p. 315
Properties at the Critical Point	p. 318
The Polyakov--Wiegmann formula	p. 319
The Affine algebra	p. 320
The WZW fields in terms of fermions	p. 322
The Sugawara form of the energy-momentum tensor	p. 323
The non-Abelian bosonization in the operator language	p. 324
Properties off the Critical Point	p. 325
Integrability of the WZNW action	p. 326
On the solution off the critical point	p. 327
Supersymmetric W ZW model	p. 329
Conclusion	p. 331
QED[subscript 2]: Operator Approach	p. 333
Introduction	p. 333
The Massless Schwinger Model	p. 335
Quantum solution	p. 335
The Maxwell current	p. 337
Chiral densities	p. 340
Vacuum structure	p. 341
Gauge transformations	p. 345
Correlation functions and violation of clustering	p. 348
Absence of charged states (screening)	p. 349
The quark-antiquark potential	p. 351
Adding flavour	p. 353
Fractional winding number and the U(1) problem	p. 356
The Massive Schwinger Model	p. 360
Equivalent bosonic formulation	p. 360
The quantum Dirac equation	p. 362
Vacuum structure and all that	p. 365
Screening versus confinement	p. 366
Adding flavour	p. 374
Lorentz transformation properties	p. 380
The MSM as the limit of a massive vector theory	p. 383
Conclusion	p. 386
Quantum Chromodynamics	p. 391
Introduction	p. 391
The 1/N expansion: 't Hooft model	p. 394
Currents, Green functions and determinants	p. 399
Tree graph expansion of the current	p. 400
Recovering the QCD[subscript 2] effective action	p. 402
Fermion Green Function	p. 405
Local decoupled formulation and BRST constraints	p. 408
Local decoupled partition function and BRST symmetries	p. 409
Systematic derivation of the constraints	p. 414
Non-local decoupled formulation and BRST constraints	p. 417
Non-local decoupled partition function and BRST symmetries	p. 417
The physical Hilbert space	p. 421
The QCD[subscript 2] vacuum	p. 422
Massive two-dimensional QCD	p. 425
Screening in two-dimensional QCD	p. 427
Further algebraic aspects	p. 433
Conclusions	p. 434
QED[subscript 2]: Functional Approach	p. 439
Introduction	p. 439
Equivalent Bosonic Action	p. 440
Gauge Invariant Correlation Functions	p. 441
The external field current, and chiral densities	p. 441
Vacuum Structure	p. 442
Chirality of the vacuum	p. 443
Why Study Gauge-Invariant Correlators	p. 447
Screening versus Confinement	p. 448
Quasi-Periodic Boundary Conditions and the [theta]-Vacuum	p. 450
Axial anomaly and the Dirac sea	p. 454
Functional Representation of Tunneling Amplitudes	p. 456
Interpretation of the Result	p. 458
Zero modes	p. 460
Calculation of det i D from the anomaly equation	p. 462
Eigenvalue Spectrum of the Dirac Operator	p. 464
Zero Modes and Boundary-Value Problem	p. 467
Free Dirac operator and non-local boundary conditions	p. 468
The little Dirac operator	p. 470
The U (1) Problem Revisited	p. 474
Conclusion	p. 479
The Finite Temperature Schwinger Model	p. 483
Introduction	p. 483
Heat kernel and Seeley expansion	p. 484
The Atiyah-Singer Index theorem	p. 488
Fermions in an Instanton potential	p. 490
Chiral condensate and symmetry breaking	p. 495
Polyakov loop-operator and screening	p. 503
Conclusion	p. 506
Non-Abelian Chiral Gauge Theories	p. 509
Introduction	p. 509
Anomalies and Cocycles	p. 514
Consistent anomaly	p. 514
More about cocycles	p. 518
Gauss anomaly	p. 520
Relation between consistent and covariant anomaly	p. 521
Isomorphic Representations of Chiral QCD[subscript 2]	p. 524
Gauge-invariant embedding	p. 525
External Field Ward Identities	p. 527
Construction of the one-Cocycle from the Anomaly	p. 533
Bosonic Action in the GNI and GI Formulation	p. 534
Symmetries of the Model	p. 538
Relation of Source Currents in GNI and GI Formulations	p. 540
Poisson Algebra of the Currents	p. 541
Hamiltonian Quantization	p. 545
Fermionization of [alpha subscript 1] [A, g]	p. 554
BRST Quantization of GI Formulation	p. 555
Chiral QCD[subscript 2] in Terms of Chiral Bosons	p. 562
Constraint Structure from the Fermionic Hamiltonian	p. 567
Chiral QCD[subscript 2] in the local decoupled formulation	p. 575
Gauge non-invariant formulation	p. 575
Gauge-invariant formulation	p. 584
Conclusion	p. 587
Chiral Quantum Electrodynamics	p. 593
Introduction	p. 593
The JR Model	p. 594
Quantization in the GNI Formulation	p. 596
Hamiltonian and constraints	p. 596
Commutation relations	p. 598
Current-potential and bosonic representation of fermion field	p. 600
Energy-momentum tensor	p. 602
Vector-field two-point function	p. 603
Fermionic two-point function	p. 604
Quantization in the GI Formulation	p. 604
Hamiltonian and constraints	p. 604
Implementation of gauge conditions	p. 606
Isomorphism between GI and GNI formulations: phase space view	p. 608
WZ term and BFT Hamiltonian embedding	p. 611
Alternative approach to quantization	p. 616
Operator solution in Lorentz-type gauges	p. 617
Path-Integral Formulation	p. 618
Perturbative Analysis in the Fermionic Formulation	p. 625
Perturbative analysis in the GNI formulation	p. 625
Perturbative analysis in the GI formulation	p. 630
Anomalous Poisson Brackets Revisited	p. 632
Operator view of anomalous Poisson brackets	p. 633
Bjorken-Johnson-Low view of anomalous Poisson brackets	p. 635
Reconstruction of commutators of the GNI formulation	p. 635
Chiral QED[subscript 2] in terms of Chiral Bosons	p. 638
Conclusion	p. 641
Conformally Invariant Field Theory	p. 645
Introduction	p. 645
Conformal transformations and conformal group	p. 646
Dilatations	p. 647
The conformal group in D dimensions	p. 647
The conformal group in two dimensions	p. 653
Mobius transformations	p. 655
The BPZ construction	p. 659
Primary and quasi-primary fields	p. 659
Radial quantization	p. 666
Descendants of primary fields	p. 671
Virasoro algebra	p. 675
Realization of Conformal Algebra for c [ 1	p. 683
Superconformal Symmetry	p. 688
Conclusion	p. 691
Conformal Field Theory with Internal Symmetry	p. 695
Introduction	p. 695
Conformal algebra and Ward identities	p. 695
Realizations of non-Abelian conformal algebra	p. 700
The Wess-Zumino-Witten field	p. 700
The non-Abelian Thirring field at the Critical Point	p. 706
Coset description of CQFT	p. 711
Coset realization of the FQS minimal unitary series	p. 712
Fermionic coset realization of SU (N)[subscript 1]	p. 713
Fermionic coset realization of FQS series	p. 716
Reduction formula for negative level WZW fields	p. 718
Critical statistical models	p. 722
Fermionic coset description of the critical Ising model	p. 722
Conclusions	p. 730
2D gravity and string related topics	p. 733
Introduction	p. 733
The Nambu-Goto string	p. 734
The effective action of 2D quantum gravity	p. 736
Uniqueness of the Polyakov action	p. 736
Quantum Gravity	p. 738
The Liouville theory	p. 746
The classical Liouville theory	p. 747
The quantum Liouville theory	p. 750
Gravity in the light-cone gauge	p. 753
Canonical quantization and SL (2, R) symmetry	p. 753
Operator product expansions and Ward identities	p. 759
Interaction of matter fields with gravity	p. 760
Two-Dimensional Supergravity	p. 762
Conclusion	p. 768
Final Remarks	p. 771
Appendices
Notation (Minkowski Space)	p. 775
Notation (Euclidean Space)	p. 781
Further Conventions	p. 785
Functional Bosonization of the Massive Thirring Model	p. 789
Bosonization of the Fermionic Kinetic Term	p. 793
Classical Integrability in the Massive Thirring Model	p. 795
Quantum Non-Local Charge: Action on Asymptotic States	p. 797
S-Matrices	p. 801
Complete S-matrix of the Gross-Neveu Model	p. 805
Poisson Brackets and Commutators	p. 809
Chiral Bosons	p. 811
Axial Anomaly from Dispersion Relations	p. 817
Loop Expansion in QCD[subscript 2]	p. 821
Index	p. 826
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Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (2nd Edition)

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Hardcover

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