Non-Perturbative Field Theory: From Two Dimensional Conformal Field Theory to QCD in Four Dimensions (Cambridge Monographs on Mathematical Physics)

"Providing a new perspective on quantum field theory, this book gives a pedagogical and up-to-date exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research work in theoretical physics. It describes in detail non-perturbative methods in quantum field theory, and explores two- dimensional and four- dimensional gauge dynamics using those methods. The book concludes with a summary emphasizing the interplay between two- and four- dimensional gauge theories. Aimed at graduate students and researchers, this book covers topics from two-dimensional conformal symmetry, affine Lie algebras, solitons, integrable models, bosonization, and 't Hooft model, to four-dimensional conformal invariance, integrability, large N expansion, Skyrme model, monopoles and instantons. Applications, first to simple field theories and gauge dynamics in two dimensions, and then to gauge theories in four dimensions and quantum chromodynamics (QCD) in particular, are thoroughly described"--Provided by publisher. "Field theory is the framework with which one describes the theory of the standard model of elementary particles and their interactions. The electromagnetic sector (QED) of the standard model is understood extremely well using perturbation theory, but the color interaction (QCD) which is responsible for hadron physics can only be accounted for perturbatively for a limited set of observational data. Due to the fact that at long distances the color interaction is strongly coupled, one cannot reliably apply perturbative methods to extract, for instance, the spectrum of the hadrons. The arsenal of tools to handle strongly coupled systems is obviously much more limited than the one used for weakly coupled ones. Nevertheless, several methods to handle non-perturbative field theories have been developed. The main goal of this book is to expose the reader to those techniques and to describe their applications in two-dimensional and four-dimensional field theories and finally in QCD in four dimensions"--Provided by publisher. Half-title......Page 3 Series-title......Page 4 Title......Page 7 Copyright......Page 8 Dedication......Page 9 Contents......Page 11 Preface......Page 17 Acknowledgements......Page 20 PART I Non-perturbative methods in two-dimensional field theory......Page 21 1.1 Complex geometry......Page 23 1.2 Free massless scalar field......Page 24 1.3 Symmetries of the classical action......Page 25 1.4 Mode expansion......Page 26 1.6 Canonical quantization......Page 27 1.7 Radial quantization......Page 29 1.8 Operator product expansion......Page 31 1.9 Path integral quantization......Page 32 1.10 Affine current algebra......Page 33 1.11 Virasoro algebra......Page 34 2.1 Conformal symmetry in two dimensions......Page 37 2.2 Primary fields......Page 38 2.3 Conformal properties of the energy-momentum tensor......Page 40 2.4 Virasoro algebra for CFT......Page 41 2.5 Descendant operators......Page 42 2.6 Hilbert space of states......Page 43 2.7 Unitary CFT and Kac determinant......Page 45 2.8 Characters......Page 48 2.9 Correlators and the conformal Ward identity......Page 49 2.10 Crossing symmetry, duality and bootstrap......Page 51 2.11 Verlinde's formula......Page 53 2.12 Free Majorana fermions – an example of a CFT......Page 54 2.13 The Ising model – the m = 3 unitary minimal model......Page 57 3.1 Simple finite-dimensional Lie algebras......Page 59 3.1.2 Cartan matrix and Dynkin diagrams......Page 61 3.1.3 Highest weight states......Page 63 3.2 Affine current algebra......Page 64 3.2.1 Cartan matrix and Dynkin diagrams......Page 66 3.2.2 The Weyl group......Page 67 3.2.3 Highest weight representations......Page 68 3.3 Current OPEs and the Sugawara construction......Page 69 3.4 Primary fields......Page 71 3.5 ALA characters......Page 72 3.6 Correlators, null vectors and the Knizhnik-Zamolodchikov equation......Page 73 3.7.1 Free Majorana fermions and.........Page 75 3.7.2 Primary fields......Page 77 3.8 Free Dirac fermions and the.........Page 78 4.1 From free massless scalar theory to the WZW model......Page 81 4.2 Perturbative conformal invariance......Page 85 4.3 ALA, Sugawara construction and the Virasoro algebra......Page 86 4.4 Correlation functions of primary fields......Page 87 4.5 WZW models with boundaries – D branes......Page 91 4.6 G/H coset models......Page 93 4.7 G/G coset models......Page 95 5.2 From the theory of a massive free scalar field to integrable models......Page 99 5.3 Classical solitons......Page 101 5.4 Breathers or "doublets"......Page 106 5.5 Quantum solitons......Page 108 5.5.1 Quantization of the breather......Page 110 5.6 Integrability and factorized S-matrix......Page 112 5.7 Yang-Baxter equations......Page 114 5.8 The general solution of the S-matrix......Page 115 5.8.1 The S-matrix of the sine-Gordon model......Page 117 5.9 From conformal field theories to integrable models......Page 119 5.10 Conserved charges and classical integrability......Page 121 5.10.1 The Lax pair method......Page 122 5.10.2 The generating function method......Page 123 5.11.1 Multilocal charges from integral equation......Page 124 5.11.2 Charges by inductive procedure......Page 126 5.12 Quantum integrable charges in the O(N) model......Page 127 5.13 Non-local charges and quantum groups......Page 128 5.14.1 The XXX1/2 model......Page 131 5.14.2 Bethe ansatz equations......Page 134 5.14.3 The thermodynamic Bethe ansatz......Page 138 5.14.4 Spin chain model in discrete time......Page 142 5.14.5 The discretized version of the sine-Gordon model......Page 143 5.15 The continuum thermodynamic Bethe ansatz......Page 145 6 Bosonization......Page 151 6.1.1 Bosonization of a free massless Dirac fermion......Page 152 6.2 Duality between the Thirring model and the sine-Gordon model......Page 156 6.3.1 Bosonization of Majorana fermions......Page 159 6.3.2 Bosonization of Dirac fermions......Page 161 6.3.3 The bosonization of a mass bilinear of Dirac fermions......Page 162 6.3.4 Bosonization of Dirac fermions with color and flavor......Page 163 6.3.5 Bosonization of mass bilinears in the product scheme......Page 166 6.3.6 Bosonization of the U(NF NC) WZW action......Page 167 6.4 Chiral bosons......Page 168 6.4.1 Chiral boson via coupling to fictitious "light-cone gravity"......Page 169 6.4.2 Non manifestly Lorentz invariant classical action......Page 172 6.4.3 Coupling to abelian gauge fields......Page 177 6.4.4 Chiral WZW and coupling to non-abelian gauge fields......Page 178 6.5 Bosonization of systems of operators of high conformal dimension......Page 179 6.5.1 The bosonization of the "b,c" free CFT......Page 180 6.5.2 The bosonization of the beta, gamma system......Page 182 6.5.3 The Wakimoto bosonization......Page 183 7.1 Introduction......Page 185 7.2 The Gross-Neveu model......Page 186 7.3 The CPN-1 model......Page 191 PART II Two-dimensional non-perturbative gauge dynamics......Page 195 8.1 Pure Maxwell theory......Page 197 8.2 QED2 – Schwinger's model......Page 198 8.3 Yang-Mills theory......Page 199 8.4 Quantum chromodynamics......Page 200 9.1 QED2 – The massive Schwinger model......Page 203 9.2 Abelian bosonization of flavored QCD2......Page 205 9.3.1 Gauging the WZW action......Page 207 9.3.2 Multiflavor QCD2 using the U(NFxNC) scheme......Page 209 10 The 't Hooft solution of 2d QCD......Page 211 10.1 Scattering of mesons......Page 218 10.2 Higher 1/N corrections......Page 221 11.2 Universality of conformal field theories coupled to YM2......Page 223 11.3.1 The basic setup......Page 226 11.3.2 't Hooft-like equation for the two-current wave function......Page 229 11.3.3 The two-current mesonic spectrum......Page 232 11.3.4 Special cases: Nf = 1, Nf = Nc and Nf >> Nc......Page 233 11.4 The adjoint vacuum and its one-current state......Page 236 11.4.1 The action of M2 on the one-current states......Page 238 12.1 Discretized light-cone quantization......Page 243 12.2 Application of DLCQ to QCD2 with fundamental fermions......Page 244 12.3 The spectrum of QCD2 with adjoint fermions......Page 248 13.1 The strong coupling limit......Page 257 13.2 Classical soliton solutions......Page 259 13.3 Semi-classical quantization and the baryons......Page 260 13.5 Quark flavor content of the baryons......Page 267 13.6 Multibaryons......Page 269 13.7 States, wave functions and binding energies......Page 270 13.8 Meson-baryon scattering......Page 272 13.8.1 Abelian case......Page 274 13.8.2 The non-abelian case......Page 276 13.8.3 Extension to arbitrary coupling......Page 279 14.1 The string tension of the massive Schwinger model......Page 285 14.3 Beyond the small mass abelian string tension......Page 288 14.4 Correction to the leading long distance abelian potential......Page 289 14.5 Finite temperature......Page 291 14.6 Two-dimensional QCD......Page 292 14.7 Symmetric and antisymmetric representations......Page 296 15.2 The action......Page 299 15.3 Two-dimensional Yang-Mills theory......Page 302 15.4 Schwinger model revisited......Page 303 15.5 Back to the YM theory......Page 305 15.6 An alternative formulation......Page 307 15.7 The resolution of the puzzle......Page 308 15.8 On bosonized QCD2......Page 309 15.9 Summary and discussion......Page 310 16.1 Introduction......Page 311 16.2 The partition function of the YM2 theory......Page 312 16.3 The partition function of gYM2 theories......Page 316 16.4 Loop averages in the generalized case......Page 317 16.5 Stringy YM2 theory......Page 319 16.6 Toward the stringy generalized YM2......Page 321 16.7 Examples......Page 322 16.8 Summary......Page 324 PART III From two to four dimensions......Page 327 17 Conformal invariance in four-dimensional field theories and in QCD......Page 329 17.1 Conformal symmetry algebra in four dimensions......Page 330 17.2 Conformal invariance of fields, Noether currents and conservation laws......Page 332 17.3 Collinear and transverse conformal transformations of fields......Page 334 17.4 Collinear primary fields and descendants......Page 336 17.5 Conformal operator product expansion......Page 338 17.6 Conformal Ward identities......Page 339 17.7 Conformal invariance and QCD4......Page 342 18 Integrability in four-dimensional gauge dynamics......Page 349 18.1 Integrability of large N four-dimensional N = 4 SYM......Page 350 18.2 High energy scattering and integrability......Page 353 19.1 Large N QCD in four dimensions......Page 357 19.1.1 Counting rules for correlation functions......Page 362 19.2 Meson phenomenology......Page 363 19.2.1 Axial U(1) and the mass of the eta'......Page 365 19.3 Baryons in the large N expansion......Page 366 19.3.1 The Hartree approximation......Page 367 19.3.2 Baryons made out of heavy quarks......Page 368 19.3.3 Baryons made out of light quarks......Page 369 19.3.4 Baryonic excited states......Page 371 19.4 Scattering processes......Page 372 20.2 The Skyrme action......Page 375 20.2.1 The Sigma term......Page 376 20.2.2 The WZ term......Page 377 20.2.3 The Skyrme term......Page 378 20.2.4 A mass term......Page 379 20.2.5 Gauging the Skyrme action......Page 380 20.3.1 The classical Skyrmion......Page 381 20.3.2 Semiclassical quantization of the soliton......Page 384 20.3.3 The Skyrme model and large Nc QCD......Page 386 20.4 The Skyrme model for Nf = 3......Page 387 21.1 Introduction......Page 391 21.2 The Yang-Mills Higgs theory - basics......Page 392 21.3 Topological solitons and magnetic monopoles......Page 393 21.4 The 't Hooft-Polyakov magnetic monopole solution......Page 396 21.5 Charge quantization......Page 397 21.6 Zero modes, time-dependent solutions and dyons......Page 398 21.7 BPS monopoles and dyons......Page 401 21.8 Montonen Olive duality......Page 402 21.9 Nahm construction of multimonopole solutions......Page 403 21.9.1 SU(2) two-monopole solutions......Page 405 21.10 Moduli space of monopoles......Page 406 22.1 The basic properties of the instanton......Page 409 22.2 The ADHM construction of instantons......Page 414 22.3 On the moduli space of instantons......Page 416 22.4 Instantons and tunneling between the vacua of the YM theory......Page 420 22.5 Instantons, theta vacua and the UA(1) anomaly......Page 423 23.1 General......Page 427 23.2 Conformal invariance......Page 428 23.3 Integrability......Page 430 23.4 Bosonization......Page 431 23.5 Topological field configurations......Page 432 23.6 Confinement versus screening......Page 434 23.7.1 Mesons......Page 436 23.7.2 Baryons......Page 438 23.8.1 Further progress in the application of the methods discussed in the book......Page 440 23.8.3 Developments in gauge dynamics due to other methods......Page 441 References......Page 443 Index......Page 453 Machine generated contents note: Preface; Acknowledgements; Part I. Non-Perturbative Methods in Two Dimensional Field Theory: 1. From massless free scalar field to conformal field theories; 2. Conformal field theory; 3. Theories invariant under affine current algebras; 4. Wess-Zumino-Witten model and Coset models; 5. Solitons and two dimensional integrable models; 6. Bosonization; 7. The large N limit of two dimensional models; Part II. Two Dimensional Non-Perturbative Gauge Dynamics: 8. Gauge theories in two dimensions - basics; 9. Bosonized gauge theories; 10. The t'Hooft solution of 2d QCD; 11. Mesonic spectrum from current algebra; 12. DLCQ and the spectra of QC with fundamental and adjoint fermions; 13. The baryonic spectrum of multiflavour QCD2 in the strong coupling limit; 14. Confinement versus screening; 15. QCD2, Coset models and BRST quantization; 16. Generalized Yang Mills theory on Riemann surface; Part III. From Two to Four Dimensions: 17. Conformal invariance in four dimensional field theories and in QCD; 18. Integrability in four dimensional gauge dynamics; 19. Large N methods in QCD4; 20. From 2d bosonized baryons to 4d skyrmions; 21. From two dimensional solitons to four dimensional magnetic monopoles; 22. Instantons of QCD; 23. Summary, conclusions and outlook; References; Index.