Non-Perturbative Methods in Two-Dimensional Quantum Field Theory

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

• ISBN13: 9789810245962
• ISBN10: 9810245963
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-11-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Table of Contents

Introduction	17	(25)
Free Fields	25	(16)
Introduction	25	(1)
Bosonic Free Fields	25	(4)
Fermionic Free Fields	29	(2)
Bosonization of Massless Fermions	31	(4)
The RS-Model	35	(3)
Conclusions	38	(3)
The Thirring model	41	(24)
Introduction	41	(1)
The Massless Thirring Model	42	(3)
The Massive Thirring Model	45	(8)
Equivalence with sine-Gordon equation	45	(2)
Classical conservation laws	47	(1)
Quantum conservation laws	48	(5)
Bosonization Revisited	53	(2)
Fermions in terms of bosons	53	(2)
The Soliton as a Disorder Parameter	55	(6)
Conclusion	61	(4)
Determinants and Heat Kernels	65	(62)
Introduction	65	(1)
Functional Determinant, one-loop diagram	66	(15)
Determinants and the Generalized Zeta-Function	69	(6)
One Point Compactification	75	(3)
The associated Dirac operator	78	(3)
Calculating Seeley Coefficients	81	(7)
The perturbative approach	81	(2)
The Schwinger-DeWitt method	83	(3)
The Fujikawa method	86	(2)
Computing Functional Determinants	88	(7)
ζ-function regularization	88	(2)
Proper-time regularization	90	(1)
The Fujikawa point of view	91	(4)
A Theorem on a one parameter family of factorizable operators	95	(3)
The QCD2 functional determinant	98	(3)
Zero-modes	101	(6)
Axial anomaly equation in the presence of zero-modes	101	(3)
Atiyah-Singer Index Theorem	104	(3)
Ambiguities in Functional Determinants	107	(4)
Ambiguities in the regularization	107	(1)
Dependence on the scale parameter	108	(3)
Mass expansion in proper-time regularization	111	(4)
The Finite Temperature Heat Kernel	115	(8)
Scalar field in a static background potential	117	(2)
Scalar field in a static background gauge potential	119	(4)
Conclusion	123	(4)
Self-Interacting fermionic models	127	(28)
Introduction	127	(1)
The O(N) Invariant Gross-Neveu Model	127	(14)
Classical conservation laws	128	(1)
Effective potential and β-function in a 1/N expansion	129	(4)
The 1/N Expansion: Feynman rules	133	(2)
Leading order S-matrix elements	135	(3)
Quantization of the non-local charge	138	(3)
Chiral Gross-Neveu Model	141	(11)
Cancellation of infrared singularities	142	(2)
The 1/N expansion	144	(2)
Operator formulation	146	(5)
Quantization of non-local charge	151	(1)
Conclusions and Physical Interpretation	152	(3)
Non-linear σ Models: Classical Aspects	155	(56)
Historical development	155	(1)
Sigma models and current algebra	156	(2)
Two-dimensional σ models: preliminaries	158	(8)
Purely Bosonic Non-linear σ Models	166	(18)
Formal developments	166	(6)
Dual symmetry and higher conservation laws	172	(10)
An explicit example: the Grassmannians	182	(2)
Non-linear σ Models with Fermions	184	(16)
Definition and properties	184	(4)
Dual symmetry and higher conservation laws	188	(7)
Construction of an explicit example	195	(5)
Analogies with 4D Gauge Theories	200	(4)
Concluding Remarks	204	(7)
Non-linear σ Models - Quantum Aspects	211	(62)
Introduction	211	(1)
Grassmannian Bosonic Models	212	(10)
1/N expansion	212	(6)
Renormalization	218	(1)
Infrared divergencies	219	(2)
Physical interpretation of the results	221	(1)
Grassmannian Models and Fermions	222	(11)
1/N expansion and Feynman rules	222	(5)
Physical interpretation of the results	227	(6)
Quantization of Higher Conservation Laws	233	(9)
Purely bosonic sigma models and anomalies	233	(6)
Fermionic interaction and anomaly cancellation	239	(3)
Algebra of non-local charges	242	(12)
Bosonic O(N)-symmetric sigma models	242	(12)
Non-local charges in the WZNW model	254	(3)
Perturbative Renormalization	257	(8)
Background Field Method	257	(6)
Parallelizable manifolds; applications to string theory	263	(2)
Anomalous Non-Linear σ Models in four dimensions	265	(1)
Conclusion	266	(7)
Exact S-matrices of 2D Models	273	(40)
Introduction	273	(7)
Consequences of higher conservation laws	273	(1)
Factorizable S-matrix	274	(4)
Fusion rules	278	(2)
Bound state scattering	280	(1)
S-matrices and Conservation Laws	280	(9)
SU (N) invariant S-matrices	280	(2)
Sine-Gordon and massive Thirring models	282	(5)
Exact S-matrix for O(N) symmetry	287	(1)
The ZN invariant S-matrix	288	(1)
Quantum Non-Local Charges and S-Matrices	289	(14)
S-matrices of purely fermionic models	289	(4)
S-matrices of non-linear sigma models	293	(10)
Boundary S-matrices	303	(4)
Further Developments	307	(1)
Conclusion	308	(5)
The Wess-Zumino-Witten Theory	313	(20)
Introduction	313	(2)
Existence of a Critical Point	315	(3)
Properties at the Critical Point	318	(7)
The Polyakov-Wiegmann formula	319	(1)
The Affine algebra	320	(2)
The WZW fields in terms of fermions	322	(1)
The Sugawara form of the energy-momentum tensor	323	(1)
The non-Abelian bosonization in the operator language	324	(1)
Properties off the Critical Point	325	(6)
Integrability of the WZNW action	326	(1)
On the solution off the critical point	327	(2)
Supersymmetric WZW model	329	(2)
Conclusion	331	(2)
QED2: Operator Approach	333	(58)
Introduction	333	(2)
The Massless Schwinger Model	335	(25)
Quantum solution	335	(2)
The Maxwell current	337	(3)
Chiral densities	340	(1)
Vacuum structure	341	(4)
Gauge transformations	345	(3)
Correlation functions and violation of clustering	348	(1)
Absence of charged states (screening)	349	(2)
The quark-antiquark potential	351	(2)
Adding flavour	353	(3)
Fractional winding number and the U(1) problem	356	(4)
The Massive Schwinger Model	360	(26)
Equivalent bosonic formulation	360	(2)
The quantum Dirac equation	362	(3)
Vacuum structure and all that	365	(1)
Screening versus confinement	366	(8)
Adding flavour	374	(6)
Lorentz transformation properties	380	(3)
The MSM as the limit of a massive vector theory	383	(3)
Conclusion	386	(5)
Quantum Chromodynamics	391	(48)
Introduction	391	(3)
The 1/N expansion: 't Hooft model	394	(5)
Currents, Green functions and determinants	399	(9)
Tree graph expansion of the current	400	(2)
Recovering the QCD2 effective action	402	(3)
Fermion Green Function	405	(3)
Local decoupled formulation and BRST constraints	408	(9)
Local decoupled partition function and BRST symmetries	409	(5)
Systematic derivation of the constraints	414	(3)
Non-local decoupled formulation and BRST constraints	417	(4)
Non-local decoupled partition function and BRST symmetries	417	(4)
The physical Hilbert space	421	(1)
The QCD2 vacuum	422	(3)
Massive two-dimensional QCD	425	(2)
Screening in two-dimensional QCD	427	(6)
Further algebraic aspects	433	(1)
Conclusions	434	(5)
QED2: Functional Approach	439	(44)
Introduction	439	(1)
Equivalent Bosonic Action	440	(1)
Gauge Invariant Correlation Functions	441	(1)
The external field current, and chiral densities	441	(1)
Vacuum Structure	442	(5)
Chirality of the vacuum	443	(4)
Why Study Gauge-Invariant Correlators	447	(1)
Screening versus Confinement	448	(2)
Quasi-Periodic Boundary Conditions and the &thetas;-Vacuum	450	(4)
Axial anomaly and the Dirac sea	454	(2)
Functional Representation of Tunneling Amplitudes	456	(2)
Interpretation of the Result	458	(6)
Zero modes	460	(2)
Calculation of det i D from the anomaly equation	462	(2)
Eigenvalue Spectrum of the Dirac Operator	464	(3)
Zero Modes and Boundary-Value Problem	467	(7)
Free Dirac operator and non-local boundary conditions	468	(2)
The little Dirac operator	470	(4)
The U(1) Problem Revisited	474	(5)
Conclusion	479	(4)
The Finite Temperature Schwinger Model	483	(26)
Introduction	483	(1)
Heat kernel and Seeley expansion	484	(4)
The Atiyah-Singer Index theorem	488	(2)
Fermions in an Instanton potential	490	(5)
Chiral condensate and symmetry breaking	495	(8)
Polyakov loop-operator and screening	503	(3)
Conclusion	506	(3)
Non-Abelian Chiral Gauge Theories	509	(84)
Introduction	509	(5)
Anomalies and Cocyles	514	(10)
Consistent anomaly	514	(4)
More about cocycles	518	(2)
Gauss anomaly	520	(1)
Relation between consistent and convariant anomaly	521	(3)
Isomorphic Representations of Chiral QCD2	524	(43)
Gauge-invariant embedding	525	(2)
External Field Ward Identities	527	(6)
Construction of the one-Cocycle from the Anomaly	533	(1)
Bosonic Action in the G N I and GI Formulation	534	(4)
Symmetries of the Model	538	(2)
Relation of Source Currents in G N I and GI Formulations	540	(1)
Poisson Algebra of the Currents	541	(4)
Hamiltonian Quantization	545	(9)
Fermionization of α1[A, g]	554	(1)
BRST Quantization of GI Formulation	555	(7)
Chiral QCD2 in Terms of Chiral Bosons	562	(5)
Constraint Structure from the Fermionic Hamiltonian	567	(8)
Chiral QCD2 in the local decoupled formulation	575	(12)
Gauge non-invariant formulation	575	(9)
Gauge-invariant formulation	584	(3)
Conclusion	587	(6)
Chiral Quantum Electrodynamics	593	(52)
Introduction	593	(1)
The JR Model	594	(2)
Quantization in the G N I Formulation	596	(8)
Hamiltonian and constraints	596	(2)
Commutation relations	598	(2)
Current-potential and bosonic representation of fermion field	600	(2)
Energy-momentum tensor	602	(1)
Vector-field two-point function	603	(1)
Fermionic two-point fucntion	604	(1)
Quantization in the GI Formulation	604	(14)
Hamiltonian and constraints	604	(2)
Implementation of gauge conditions	606	(2)
Isomorphism between GI and G N I formulations: phase space view	608	(3)
WZ term and BFT Hamiltonian embedding	611	(5)
Alternative approach to quantization	616	(1)
Operator solution in Lorentz-type gauges	617	(1)
Path-Integral Formulation	618	(7)
Perturbative Analysis in the Fermionic Formulation	625	(7)
Perturbative analysis in the G N I formulation	625	(5)
Perturbative analysis in the GI formulation	630	(2)
Anomalous Poisson Brackets Revisited	632	(6)
Operator view of anomalous Poisson brackets	633	(2)
Bjorken-Johnson-Low view of anomalous Poisson brackets	635	(1)
Reconstruction of commutators of the G N I formulation	635	(3)
Chiral QED2 in terms of Chiral Bosons	638	(3)
Conclusion	641	(4)
Conformally Invariant Field Theory	645	(50)
Introduction	645	(1)
Conformal transformations and conformal group	646	(7)
Dilatations	647	(1)
The conformal group in D dimensions	647	(6)
The conformal group in two dimensions	653	(6)
Mobius transformations	655	(4)
The BPZ construction	659	(24)
Primary and quasi-primary fields	659	(7)
Radial quantization	666	(5)
Descendants of primary fields	671	(4)
Virasoro algebra	675	(8)
Realization of Conformal Algebra for c < 1.	683	(5)
Superconformal Symmetry	688	(3)
Conclusion	691	(4)
Conformal Field Theory with Internal Symmetry	695	(38)
Introduction	695	(1)
Conformal algebra and Ward identities	695	(5)
Realizations of non-Abelian conformal algebra	700	(11)
The Wess-Zumino-Witten field	700	(6)
The non-Abelian Thirring field at the Critical Point	706	(5)
Coset description of CQFT	711	(11)
Coset realization of the FQS minimal unitary series	712	(1)
Fermionic coset realization of SU (N)1	713	(3)
Fermionic coset realization of FQS series	716	(2)
Reduction formula for negative level WZW fields	718	(4)
Critical statistical models	722	(8)
Fermionic coset description of the critical Ising model	722	(8)
Conclusions	730	(3)
2D gravity and string related topics	733	(38)
Introduction	733	(1)
The Nambu-Goto string	734	(2)
The effective action of 2D quantum gravity	736	(10)
Uniqueness of the Polyakov action	736	(2)
Quantum Gravity	738	(8)
The Liouville theory	746	(7)
The classical Liouville theory	747	(3)
The quantum Liouville theory	750	(3)
Gravity in the light-cone gauge	753	(15)
Canonical quantization and SL(2, R) symmetry	753	(6)
Operator product expansions and Ward identities	759	(1)
Interaction of matter fields with gravity	760	(2)
Two-Dimensional Supergravity	762	(6)
Conclusion	768	(3)
Final Remarks	771	(55)
Appendices
A Notation (Minkowski Space)	775	(6)
B Notation (Euclidean Space)	781	(4)
C Further Conventions	785	(4)
D Functional Bosonization of the Massive Thirring Model	789	(4)
E Bosonization of the Fermionic Kinetic Term	793	(2)
F Classical Integrability in the Massive Thirring Model	795	(2)
G Quantum Non-Local Charge: Action on Asymptotic States	797	(4)
H S-Matrices	801	(4)
I Complete S-matrix of the Gross-Neveu Model	805	(4)
J Poisson Brackets and Commutators	809	(2)
K Chiral Bosons	811	(6)
L Axial Anomaly from Dispersion Relations	817	(4)
M Loop Expansion in QCD2	821	(5)
Index	826