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Ќазвание: Solitons and instantons
јвтор: Rajaraman R.
This book offers an elementary and unified introduction to the non-perturbative results obtained in relativistic quantum field theory based on classical soliton and instanton solutions. Such solutions are derived for a variety of models and classified by topological indices. The methods are then developed for quantizing solitons to obtain quantum particles. Vacuum tunneling, &ugr;-vacua and the dilute-instanton-gas approximation are described in detail. Other instanton effects related to quark-quark forces, confinement, the U(1) problem and Borel summability are also discussed. The emphasis is on presenting the basic ideas in a simple pedagogical way. Technical tools like functional methods, Grassman integrals, homotopy classification, collective co-ordinates etc. are developed ab initio.
The presentation of this work is kept at a fairly simple level and ideas are developed through illustrative examples. Techniques not covered in older field theory textbooks, such as functional integral methods, are presented in some detail to the necessary extent. These techniques are important in their own right. Although the book is mainly addressed to particle physicists and quantum field theorists, several portions will be of relevance to other branches of physics, particularly statistical mechanics. These include three chapters devoted to deriving classical soliton and instanton solutions and one on collective co-ordinates, as well as sections devoted to general techniques
"Classical limit" of Fermi fields264Ч265272 "Classical limit" of Fermi fields, functional integrals over272Ч273 "Elementary" bosons208 209 "Elementary" bosons, disappearance of216 model116 model, gauge invariance in117 model, instantons of122 model, relationship to O(3) model123 -vacua and -sectors333Ч337341Ч343 -vacua and -sectors and cluster decomposition371Ч372 -vacua and -sectors, background field in352Ч353372Ч373 -vacua and -sectors, degeneracy and chiral symmetry372 -vacua and -sectors, restoration of equivalence of373 'tHooft Ч Polyakov monopoles and the Bogomol'nyi condition69Ч70 'tHooft Ч Polyakov monopoles and the Dirac Ч Schwinger quantisation condition67 'tHooft Ч Polyakov monopoles, as solitary waves70 'tHooft Ч Polyakov monopoles, force between83 'tHooft Ч Polyakov monopoles, generalisation to higher groups and other representations73Ч74 'tHooft Ч Polyakov monopoles, homotopy analysis62Ч63 'tHooft Ч Polyakov monopoles, mass of70 'tHooft Ч Polyakov monopoles, topological current and charge64Ч65 1/N expansion115349384 Abelian (Higgs) gauge model75113318 Abelian (Higgs) gauge model, -vacua in333Ч337 Abelian (Higgs) gauge model, confinement in356Ч357 Abelian (Higgs) gauge model, in (2 + 1) dimensions76 Abelian (Higgs) gauge model, instantons in (1 + 1) dimensions114331 Abelian (Higgs) gauge model, on Higgs phenomenon in318Ч319 Abelian (Higgs) gauge model, solitons of77Ч78 Abelian (Higgs) gauge model, topological |N> vacua in325 Abelian (Higgs) gauge model, vacuum tunnelling in331Ч332 Action-angle variables41Ч42200 Analyticity225Ч226 anomalies78see Anti-commuting c-numberssee "Grassmann numbers" Anti-instanton of PPP306 Anti-instanton of the Yang Ч Mills system104 asymptotic freedom8 Asymptotic freedom of QCD359 Asymptotic freedom of the O(3) model115 Asymptotic freedom of the Yang Ч Mills system347 Axial current anomaly361364Ч365 Axial current anomaly and vacuum tunnelling suppression365Ч370 Axions383 Backlund transformations643Ч44 Backlund transformations for the sine-Gordon system43Ч44 Binding energy of SG bound states209Ч211 Binding energy of SG bound states by Bethe Ч Salpeter equation210Ч211 Binding energy of SG bound states, non-relativistic limit210 Bloch's theorem305332 Bloch's theorem, derivation using instantons312 Bogomolnyi condition69Ч70 Bohr Ч Sommerfeld condition, derivation using path integrals183Ч187 Bohr Ч Sommerfeld condition, generalisation in field theory195 Borel function375 Borel function for GreenТs functions376Ч377 Borel sum375 Borel sum, generalisation379 Borel summation procedure374Ч375 Borel summation procedure, impact of instantons on377Ч380 Bosonisation217 Breather solutions of the sine-Gordon modelsee "Doublet solutions" CDD ambiguities232 CDD ambiguities, removal of, for O(N) and GN models235Ч236 CDD ambiguities, removal of, in the SG model233Ч235 Charge conjugation279295Ч297 Charge conjugation, matrices279295 Charge conjugation, operator296Ч297 Charge-monopole duality74 Charged soliton solutions27249Ч252 Chiral SU(2) SU(2) symmetry360 Chiral SU(2) SU(2) symmetry and massless quarks360 Chiral SU(2) SU(2) symmetry, spontaneous breaking of360Ч361 Chiral U(1) charge and suppression of tunnelling370 Chiral U(1) charge, gauge invariant ()369 Chiral U(1) charge, gauge variant ()369 Chirality of operators370 Cluster decomposition370Ч372 Collective coordinates237 Collective coordinates for general symmetries262 Collective coordinates for translation symmetry254 Collective coordinates in a U(1) symmetric model246Ч247 Collective coordinates in Euclidean functional integral390Ч393 Collective coordinates in non-relativistic quantum mechanics238Ч243 Collective coordinates, Jacobian associated with393 Colour group358 Colour group, indices of359 Colour group, suppression of indices of360363 confinement3 Confinement and instantons in QCD357Ч358 Confinement of charges by instantons356Ч357 Confinement of quarks in QCD3348351 Continuous symmetry and the semi-classical method131150Ч153 Continuous symmetry in a complex scalar field theory244 Continuous symmetry in a non-relativistic problem238 Crossing symmetry225 Crossing symmetry for (1 + 1) dimensional O(N) models227 Cubic identities for S matrices223 Cubic identities for S matrices for O(N) models227Ч228 Dilute instanton gas313331 Dilute instanton gas, difficulties with, in Yang Ч Mills theory344Ч347 Dilute instanton gas, self consistency of, in PPP313 Dirac field theory with quartic interaction280299Ч301 Dirac field theory, energy spectrum for free field277 Dirac field theory, functional integral for free field273Ч274 Dirac field theory, functional integral for interacting field280 Dirac Hamiltonian (single particle)275278285 Dirac matrices274 Dirac matrices in (1 + 1) dimensions291 Dirac matrices, a representation of279 Doublet (or Breather) solutions, as bound states209 Doublet (or Breather) solutions, as poles in the S-matrix234 Doublet (or Breather) solutions, classical40 Doublet (or Breather) solutions, interpretation of208Ч213 Doublet (or Breather) solutions, quantisation of203 208 Doublet (or Breather) solutions, quantum mass of207 Doublet (or Breather) solutions, quantum statility of211Ч212 Dyons71Ч72 Energy band in a periodic potential304Ч305 Energy band in a periodic potential, derivation using instantons305Ч311 Euclidean action definition85 Euclidean action definition and static energy functionals112Ч113 Euclidean action definition for the model118 Euclidean action definition for the abelian Higgs model326 Euclidean action definition for the Klein Ч Gordon system85 Euclidean action definition for the pendulum112 Euclidean action definition for the Yang Ч Mills system88 Euclidean action definition of the sine-Gordon system317 Euclidean systems, action of85Ч86 Euclidean systems, definition84Ч85 Euclidean systems, field equations of8588118 Euclidean systems, functional integrals for175181Ч183 Exact S-matrices225Ч236 Exact S-matrices for the Gross Ч Neveu model236 Exact S-matrices for the non-linear O(3) model236 Exact S-matrices for the SG model233Ч235 Factorisation of S-matrices223 Fermi fields, "classical limit" of264Ч265 Fermi fields, functional integrals for272Ч273 Fermi fields, soliton quantisation for284Ч298 Fermions from Bose fields73214218Ч219 Flavour359 Floquet indices287 Flux quantisation in superconductors76Ч77 Form factors of soliton states159Ч160 Functional integrals in field theory176 Functional integrals in field theory and quantisation of static solitons178Ч180 Functional integrals in field theory and quantisation periodic solitons194Ч195 Functional integrals in field theory for abelian Higgs model329 Functional integrals in field theory for Fermi fields272Ч273280