Non-perturbative methods in 2 dimensional quantum field theory / Elcio Abdalla, M. Cristina B. Abdalla, Klaus D. Rothe.

Format:

Book

Author/Creator:

Abdalla, Elcio.

Contributor:

Abdalla, M. Cristina B.

Rothe, Klaus D. (Klaus Dieter)

Kenneth H. and Thelma F. Cisney Memorial Fund.

Language:

English

Subjects (All):

Quantum field theory--Mathematical models.

Quantum field theory.

Physical Description:

832 pages : illustrations ; 26 cm

Edition:

Second edition.

Other Title:

2 dimensional quantum field theory

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, [2001]

Summary:

Aimed at graduate students and post-doctoral researchers, this text presents a detailed survey of the developments in two-dimensional quantum field theory since the pioneering work of Thirring. Introductory chapters discuss generalized free fields and their application to the solution of two exactly solvable models. Other topics include, for example, the Gross- Neveu model, non-linear sigma models, non-Abelian fermion-boson equivalences, and conformal field theory. The second edition also features a new chapter on the finite temperature Schwinger model. Annotation copyrighted by Book News Inc., Portland, OR.

Contents:

2 Free Fields 25

2.2 Bosonic Free Fields 25

2.3 Fermionic Free Fields 29

2.4 Bosonization of Massless Fermions 31

2.5 The RS-Model 35

3 The Thirring model 41

3.2 The Massless Thirring Model 42

3.3 The Massive Thirring Model 45

3.3.1 Equivalence with sine-Gordon equation 45

3.3.2 Classical conservation laws 47

3.3.3 Quantum conservation laws 48

3.4 Bosonization Revisited 53

3.4.1 Fermions in terms of bosons 53

3.5 The Soliton as a Disorder Parameter 55

4 Determinants and Heat Kernels 65

4.2 Functional Determinant, one-loop diagram 66

4.2.1 Determinants and the Generalized Zeta-Function 69

4.2.2 One Point Compactification 75

4.2.3 The associated Dirac operator 78

4.3 Calculating Seeley Coefficients 81

4.3.1 The perturbative approach 81

4.3.2 The Schwinger-DeWitt method 83

4.3.3 The Fujikawa method 86

4.4 Computing Functional Determinants 88

4.4.1 [zeta]-function regularization 88

4.4.2 Proper-time regularization 90

4.4.3 The Fujikawa point of view 91

4.5 A Theorem on a one parameter family of factorizable operators 95

4.6 The QCD[subscript 2] functional determinant 98

4.7 Zero-modes 101

4.7.1 Axial anomaly equation in the presence of zero-modes 101

4.7.2 Atiyah-Singer Index Theorem 104

4.8 Ambiguities in Functional Determinants 107

4.8.1 Ambiguities in the regularization 107

4.8.2 Dependence on the scale parameter 108

4.9 Mass expansion in proper-time regularization 111

4.10 The Finite Temperature Heat Kernel 115

4.10.1 Scalar field in a static background potential 117

4.10.2 Scalar field in a static background gauge potential 119

5 Self-Interacting fermionic models 127

5.2 The O(N) Invariant Gross

Neveu Model 127

5.2.1 Classical conservation laws 128

5.2.2 Effective potential and [beta]-function in a 1/N expansion 129

5.2.3 The 1/N Expansion: Feynman rules 133

5.2.4 Leading order S-matrix elements 135

5.2.5 Quantization of the non-local charge 138

5.3 Chiral Gross

Neveu Model 141

5.3.1 Cancellation of infrared singularities 142

5.3.2 The 1/N expansion 144

5.3.3 Operator formulation 146

5.3.4 Quantization of non-local charge 151

5.4 Conclusions and Physical Interpretation 152

6 Non-linear [sigma] Models: Classical Aspects 155

6.1 Historical development 155

6.2 Sigma models and current algebra 156

6.3 Two-dimensional [sigma] models: preliminaries 158

6.4 Purely Bosonic Non-linear [sigma] Models 166

6.4.1 Formal developments 166

6.4.2 Dual symmetry and higher conservation laws 172

6.4.3 An explicit example: the Grassmannians 182

6.5 Non-linear [sigma] Models with Fermions 184

6.5.1 Definition and properties 184

6.5.2 Dual symmetry and higher conservation laws 188

6.5.3 Construction of an explicit example 195

6.6 Analogies with 4D Gauge Theories 200

7 Non-linear [sigma] Models - Quantum Aspects 211

7.2 Grassmannian Bosonic Models 212

7.2.1 1/N expansion 212

7.2.2 Renormalization 218

7.2.3 Infrared divergencies 219

7.2.4 Physical interpretation of the results 221

7.3 Grassmannian Models and Fermions 222

7.3.1 1/N expansion and Feynman rules 222

7.3.2 Physical interpretation of the results 227

7.4 Quantization of Higher Conservation Laws 233

7.4.1 Purely bosonic sigma models and anomalies 233

7.4.2 Fermionic interaction and anomaly cancellation 239

7.5 Algebra of non-local charges 242

7.5.1 Bosonic O(N)-symmetric sigma models 242

7.6 Non-local charges in the WZNW model 254

7.7 Perturbative Renormalization 257

7.7.1 Background Field Method 257

7.7.2 Parallelizable manifolds; applications to string theory 263

7.8 Anomalous Non-Linear [sigma] Models in four dimensions 265

8 Exact S-matrices of 2D Models 273

8.1.1 Consequences of higher conservation laws 273

8.1.2 Factorizable S-matrix 274

8.1.3 Fusion rules 278

8.1.4 Bound state scattering 280

8.2 S-matrices and Conservation Laws 280

8.2.1 SU(N) invariant S-matrices 280

8.2.2 Sine-Gordon and massive Thirring models 282

8.2.3 Exact S-matrix for O(N) symmetry 287

8.2.4 The Z[subscript N] invariant S-matrix 288

8.3 Quantum Non-Local Charges and S-Matrices 289

8.3.1 S-matrices of purely fermionic models 289

8.3.2 S-matrices of non-linear sigma models 293

8.4 Boundary S-matrices 303

8.5 Further Developments 307

9 The Wess

Zumino

Witten Theory 313

9.2 Existence of a Critical Point 315

9.3 Properties at the Critical Point 318

9.3.1 The Polyakov

Wiegmann formula 319

9.3.2 The Affine algebra 320

9.3.3 The WZW fields in terms of fermions 322

9.3.4 The Sugawara form of the energy-momentum tensor 323

9.3.5 The non-Abelian bosonization in the operator language 324

9.4 Properties off the Critical Point 325

9.4.1 Integrability of the WZNW action 326

9.4.2 On the solution off the critical point 327

9.4.3 Supersymmetric W ZW model 329

10 QED[subscript 2]: Operator Approach 333

10.2 The Massless Schwinger Model 335

10.2.1 Quantum solution 335

10.2.2 The Maxwell current 337

10.2.3 Chiral densities 340

10.2.4 Vacuum structure 341

10.2.5 Gauge transformations 345

10.2.6 Correlation functions and violation of clustering 348

10.2.7 Absence of charged states (screening) 349

10.2.8 The quark-antiquark potential 351

10.2.9 Adding flavour 353

10.2.10 Fractional winding number and the U(1) problem 356

10.3 The Massive Schwinger Model 360

10.3.1 Equivalent bosonic formulation 360

10.3.2 The quantum Dirac equation 362

10.3.3 Vacuum structure and all that 365

10.3.4 Screening versus confinement 366

10.3.5 Adding flavour 374

10.3.6 Lorentz transformation properties 380

10.3.7 The MSM as the limit of a massive vector theory 383

11 Quantum Chromodynamics 391

11.2 The 1/N expansion: 't Hooft model 394

11.3 Currents, Green functions and determinants 399

11.3.1 Tree graph expansion of the current 400

11.3.2 Recovering the QCD[subscript 2] effective action 402

11.3.3 Fermion Green Function 405

11.4 Local decoupled formulation and BRST constraints 408

11.4.1 Local decoupled partition function and BRST symmetries 409

11.4.2 Systematic derivation of the constraints 414

11.5 Non-local decoupled formulation and BRST constraints 417

11.5.1 Non-local decoupled partition function and BRST symmetries 417

11.6 The physical Hilbert space 421

11.7 The QCD[subscript 2] vacuum 422

11.8 Massive two-dimensional QCD 425

11.9 Screening in two-dimensional QCD 427

11.10 Further algebraic aspects 433

12 QED[subscript 2]: Functional Approach 439

12.2 Equivalent Bosonic Action 440

12.3 Gauge Invariant Correlation Functions 441

12.3.1 The external field current, and chiral densities 441

12.4 Vacuum Structure 442

12.4.1 Chirality of the vacuum 443

12.5 Why Study Gauge-Invariant Correlators 447

12.6 Screening versus Confinement 448

12.7 Quasi-Periodic Boundary Conditions and the [theta]-Vacuum 450

12.8 Axial anomaly and the Dirac sea 454

12.9 Functional Representation of Tunneling Amplitudes 456

12.10 Interpretation of the Result 458

12.10.1 Zero modes 460

12.10.2 Calculation of det i D from the anomaly equation 462

12.11 Eigenvalue Spectrum of the Dirac Operator 464

12.12 Zero Modes and Boundary-Value Problem 467

12.12.1 Free Dirac operator and non-local boundary conditions 468

12.12.2 The little Dirac operator 470

12.13 The U (1) Problem Revisited 474

13 The Finite Temperature Schwinger Model 483

13.2 Heat kernel and Seeley expansion 484

13.3 The Atiyah-Singer Index theorem 488

13.4 Fermions in an Instanton potential 490

13.5 Chiral condensate and symmetry breaking 495

13.6 Polyakov loop-operator and screening 503

14 Non-Abelian Chiral Gauge Theories 509

14.2 Anomalies and Cocycles 514

14.2.1 Consistent anomaly 514

14.2.2 More about cocycles 518

14.2.3 Gauss anomaly 520

14.2.4 Relation between consistent and covariant anomaly 521

14.3 Isomorphic Representations of Chiral QCD[subscript 2] 524

14.3.1 Gauge-invariant embedding 525

14.3.2 External Field Ward Identities 527

14.3.3 Construction of the one-Cocycle from the Anomaly 533

14.3.4 Bosonic Action in the GNI and GI Formulation 534

14.3.5 Symmetries of the Model 538

14.3.6 Relation of Source Currents in GNI and GI Formulations 540

14.3.7 Poisson Algebra of the Currents 541

14.3.8 Hamiltonian Quantization 545

14.3.9 Fermionization of [alpha subscript 1] [A, g] 554

14.3.10 BRST Quantization of GI Formulation 555

14.3.11 Chiral QCD[subscript 2] in Terms of Chiral Bosons 562

14.4 Constraint Structure from the Fermionic Hamiltonian 567

14.5 Chiral QCD[subscript 2] in the local decoupled formulation 575

14.5.1 Gauge non-invariant formulation 575

14.5.2 Gauge-invariant formulation 584

15 Chiral Quantum Electrodynamics 593

15.2 The JR Model 594

15.3 Quantization in the GNI Formulation 596

15.3.1 Hamiltonian and constraints 596

15.3.2 Commutation relations 598

15.3.3 Current-potential and bosonic representation of fermion field 600

15.3.4 Energy-momentum tensor 602

15.3.5 Vector-field two-point function 603

15.3.6 Fermionic two-point function 604

15.4 Quantization in the GI Formulation 604

15.4.1 Hamiltonian and constraints 604

15.4.2 Implementation of gauge conditions 606

15.4.3 Isomorphism between GI and GNI formulations: phase space view 608

15.4.4 WZ term and BFT Hamiltonian embedding 611

15.4.5 Alternative approach to quantization 616

15.4.6 Operator solution in Lorentz-type gauges 617

15.5 Path-Integral Formulation 618

15.6 Perturbative Analysis in the Fermionic Formulation 625

15.6.1 Perturbative analysis in the GNI formulation 625

15.6.2 Perturbative analysis in the GI formulation 630

15.7 Anomalous Poisson Brackets Revisited 632

15.7.1 Operator view of anomalous Poisson brackets 633

15.7.2 Bjorken-Johnson-Low view of anomalous Poisson brackets 635

15.7.3 Reconstruction of commutators of the GNI formulation 635

15.8 Chiral QED[subscript 2] in terms of Chiral Bosons 638

16 Conformally Invariant Field Theory 645

16.2 Conformal transformations and conformal group 646

16.2.1 Dilatations 647

16.2.2 The conformal group in D dimensions 647

16.3 The conformal group in two dimensions 653

16.3.1 Mobius transformations 655

16.4 The BPZ construction 659

16.4.1 Primary and quasi-primary fields 659

16.4.2 Radial quantization 666

16.4.3 Descendants of primary fields 671

16.4.4 Virasoro algebra 675

16.5 Realization of Conformal Algebra for c < 1 683

16.6 Superconformal Symmetry 688

17 Conformal Field Theory with Internal Symmetry 695

17.2 Conformal algebra and Ward identities 695

17.3 Realizations of non-Abelian conformal algebra 700

17.3.1 The Wess-Zumino-Witten field 700

17.3.2 The non-Abelian Thirring field at the Critical Point 706

17.4 Coset description of CQFT 711

17.4.1 Coset realization of the FQS minimal unitary series 712

17.4.2 Fermionic coset realization of SU (N)[subscript 1] 713

17.4.3 Fermionic coset realization of FQS series 716

17.4.4 Reduction formula for negative level WZW fields 718

17.5 Critical statistical models 722

17.5.1 Fermionic coset description of the critical Ising model 722

18 2D gravity and string related topics 733

18.2 The Nambu-Goto string 734

18.3 The effective action of 2D quantum gravity 736

18.3.1 Uniqueness of the Polyakov action 736

18.3.2 Quantum Gravity 738

18.4 The Liouville theory 746

18.4.1 The classical Liouville theory 747

18.4.2 The quantum Liouville theory 750

18.5 Gravity in the light-cone gauge 753

18.5.1 Canonical quantization and SL (2, R) symmetry 753

18.5.2 Operator product expansions and Ward identities 759

18.5.3 Interaction of matter fields with gravity 760

18.5.4 Two-Dimensional Supergravity 762

A Notation (Minkowski Space) 775

B Notation (Euclidean Space) 781

C Further Conventions 785

D Functional Bosonization of the Massive Thirring Model 789

E Bosonization of the Fermionic Kinetic Term 793

F Classical Integrability in the Massive Thirring Model 795

G Quantum Non-Local Charge: Action on Asymptotic States 797

H S-Matrices 801

I Complete S-matrix of the Gross-Neveu Model 805

J Poisson Brackets and Commutators 809

K Chiral Bosons 811

L Axial Anomaly from Dispersion Relations 817

M Loop Expansion in QCD[subscript 2] 821.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Kenneth H. and Thelma F. Cisney Memorial Fund.

ISBN:

9810245963

OCLC:

48256841