Non Perturbative Methods in Two Dimensional Quantum Field Theory

by Abdalla, Elcio; Abdalla, M. Cristina B.; Rothe, Klaus D.

• ISBN13: 9789810204624
• ISBN10: 9810204620
• Format: Hardcover
• Copyright: 1991-07-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

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