Íàøëè îïå÷àòêó? Âûäåëèòå åå ìûøêîé è íàæìèòå Ctrl+Enter
Íàçâàíèå: Solitons and instantons
Àâòîð: Rajaraman R.
Àííîòàöèÿ:
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 fields264—265272 "Classical limit" of Fermi fields, functional integrals over272—273 "Elementary" bosons208 209 "Elementary" bosons, disappearance of216 model116 model, gauge invariance in117 model, instantons of122 model, relationship to O(3) model123 -vacua and -sectors333—337341—343 -vacua and -sectors and cluster decomposition371—372 -vacua and -sectors, background field in352—353372—373 -vacua and -sectors, degeneracy and chiral symmetry372 -vacua and -sectors, restoration of equivalence of373 'tHooft — Polyakov monopoles and the Bogomol'nyi condition69—70 'tHooft — Polyakov monopoles and the Dirac — Schwinger quantisation condition67 'tHooft — Polyakov monopoles, as solitary waves70 'tHooft — Polyakov monopoles, force between83 'tHooft — Polyakov monopoles, generalisation to higher groups and other representations73—74 'tHooft — Polyakov monopoles, homotopy analysis62—63 'tHooft — Polyakov monopoles, mass of70 'tHooft — Polyakov monopoles, topological current and charge64—65 1/N expansion115349384 Abelian (Higgs) gauge model75113318 Abelian (Higgs) gauge model, -vacua in333—337 Abelian (Higgs) gauge model, confinement in356—357 Abelian (Higgs) gauge model, in (2 + 1) dimensions76 Abelian (Higgs) gauge model, instantons in (1 + 1) dimensions114331 Abelian (Higgs) gauge model, on Higgs phenomenon in318—319 Abelian (Higgs) gauge model, solitons of77—78 Abelian (Higgs) gauge model, topological |N> vacua in325 Abelian (Higgs) gauge model, vacuum tunnelling in331—332 Action-angle variables41—42200 Analyticity225—226 anomalies78see Anti-commuting c-numberssee "Grassmann numbers" Anti-instanton of PPP306 Anti-instanton of the Yang — Mills system104 asymptotic freedom8 Asymptotic freedom of QCD359 Asymptotic freedom of the O(3) model115 Asymptotic freedom of the Yang — Mills system347 Axial current anomaly361364—365 Axial current anomaly and vacuum tunnelling suppression365—370 Axions383 Backlund transformations643—44 Backlund transformations for the sine-Gordon system43—44 Binding energy of SG bound states209—211 Binding energy of SG bound states by Bethe — Salpeter equation210—211 Binding energy of SG bound states, non-relativistic limit210 Bloch's theorem305332 Bloch's theorem, derivation using instantons312 Bogomolnyi condition69—70 Bohr — Sommerfeld condition, derivation using path integrals183—187 Bohr — Sommerfeld condition, generalisation in field theory195 Borel function375 Borel function for Green’s functions376—377 Borel sum375 Borel sum, generalisation379 Borel summation procedure374—375 Borel summation procedure, impact of instantons on377—380 Bosonisation217 Breather solutions of the sine-Gordon modelsee "Doublet solutions" CDD ambiguities232 CDD ambiguities, removal of, for O(N) and GN models235—236 CDD ambiguities, removal of, in the SG model233—235 Charge conjugation279295—297 Charge conjugation, matrices279295 Charge conjugation, operator296—297 Charge-monopole duality74 Charged soliton solutions27249—252 Chiral SU(2) SU(2) symmetry360 Chiral SU(2) SU(2) symmetry and massless quarks360 Chiral SU(2) SU(2) symmetry, spontaneous breaking of360—361 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 decomposition370—372 Collective coordinates237 Collective coordinates for general symmetries262 Collective coordinates for translation symmetry254 Collective coordinates in a U(1) symmetric model246—247 Collective coordinates in Euclidean functional integral390—393 Collective coordinates in non-relativistic quantum mechanics238—243 Collective coordinates, Jacobian associated with393 Colour group358 Colour group, indices of359 Colour group, suppression of indices of360363 confinement3 Confinement and instantons in QCD357—358 Confinement of charges by instantons356—357 Confinement of quarks in QCD3348351 Continuous symmetry and the semi-classical method131150—153 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) models227—228 Dilute instanton gas313331 Dilute instanton gas, difficulties with, in Yang — Mills theory344—347 Dilute instanton gas, self consistency of, in PPP313 Dirac field theory with quartic interaction280299—301 Dirac field theory, energy spectrum for free field277 Dirac field theory, functional integral for free field273—274 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 of208—213 Doublet (or Breather) solutions, quantisation of203 208 Doublet (or Breather) solutions, quantum mass of207 Doublet (or Breather) solutions, quantum statility of211—212 Dyons71—72 Energy band in a periodic potential304—305 Energy band in a periodic potential, derivation using instantons305—311 Euclidean action definition85 Euclidean action definition and static energy functionals112—113 Euclidean action definition for the model118 Euclidean action definition for the abelian Higgs model326 Euclidean action definition for the Klein — Gordon system85 Euclidean action definition for the pendulum112 Euclidean action definition for the Yang — Mills system88 Euclidean action definition of the sine-Gordon system317 Euclidean systems, action of85—86 Euclidean systems, definition84—85 Euclidean systems, field equations of8588118 Euclidean systems, functional integrals for175181—183 Exact S-matrices225—236 Exact S-matrices for the Gross — Neveu model236 Exact S-matrices for the non-linear O(3) model236 Exact S-matrices for the SG model233—235 Factorisation of S-matrices223 Fermi fields, "classical limit" of264—265 Fermi fields, functional integrals for272—273 Fermi fields, soliton quantisation for284—298 Fermions from Bose fields73214218—219 Flavour359 Floquet indices287 Flux quantisation in superconductors76—77 Form factors of soliton states159—160 Functional integrals in field theory176 Functional integrals in field theory and quantisation of static solitons178—180 Functional integrals in field theory and quantisation periodic solitons194—195 Functional integrals in field theory for abelian Higgs model329 Functional integrals in field theory for Fermi fields272—273280
Functional integrals in field theory for the Yang — Mills theory342 Functional integrals in field theory, boundary conditions177 Functional integrals in field theory, Euclidean case181—182 Functional integrals in field theory, gaussian approximation to181 Functional integrals in field theory, stationary phase approximation (SPA) to179 Gauge Fields60 Gauge fields, matrix notation for87359 Gauge fixing terms329342 Gauge transformations, abelian117—118319 Gauge transformations, non-abelian6087337 Gauge transformations, small and large323 Gauge transformations, time-independent323338 Gauss' theorem and gauge equivalence, abelian321—323 Gauss' theorem and gauge equivalence, non-abelian339—340 Gaussian approximation181306 Goldstone bosons360151 Goldstone bosons and SU(2) SU(2) symmetry360 Goldstone bosons, absence of, for U(1) symmetry361 Goldstone bosons, axions as383 Grassmann algebra, definition266 Grassmann algebra, generators of266 Grassmann algebra, odd and even subsets of266 Grassmann fields265271—272 Grassmann fields, functional integrals over272—273 Grassmann numbers265 Grassmann numbers, change of variables in integration269 Grassmann numbers, derivatives with respect to267 Grassmann numbers, exponential integral270 Grassmann numbers, integration over268 Gross — Neveu (GN) model225236 Gross — Neveu (GN) model, definition of300 Gross — Neveu (GN) model, S-matrix of236 Gross — Neveu (GN) model, semi-classical method for301 Group measure for SU(2)93—95 Half-integral charge, solitons with295—298 Heavy quark potential381 Higgs (scalar) fields, definition60 Higgs mechanismsee "Higgs phenomenon" Higgs phenomenon318—319 Higgs phenomenon, prevention by instantons320337352357 Homotopy classificationsee "Topological charge" Homotopy classification and gauge equivalence325340 Homotopy classification in liquid crystals79 Homotopy classification in the model119 Homotopy classification in the abelian Higgs model76 Homotopy classification in the Euclidean Yang — Mills system90—98 Homotopy classification in the O(3) model53—54 Homotopy classification of classical gauge vacua324—325338 Homotopy classification of mappings of circles into circles51—52 Homotopy classification of monopole solutions62—63 Homotopy classification, some general results74 Infra-red divergence in Yang — Mills instanton effects344—347 Instantons of SU(N) gauge theories114 Instantons of the model122 Instantons of the abelian Higgs model113—114 Instantons of the Yang — Mills system88103 Instantons, absence of, in O(N) models113 Instantons, general definition of284—85 Instantons, relevance of finite size of308344—345 Internal space49 Internal space, centrifugal effect in252 Internal space, global rotations in49—61 Internal space, local transformations in59—61 Internal space, rotation in249252 Inverse scattering method6915 Inverse scattering method for the sine-Gordon system41—42 Kink of - theory, anti-kink20 Kink of - theory, classical solution20 Kink of - theory, fluctuation modes about140—141 Kink of - theory, form factors of159—160 Kink of - theory, mass of classical solution21 Kink of - theory, mass of the quantum kink144—148 Kink of - theory, non-perturbative nature23 Kink of - theory, quantisation of137—150 Kink of - theory, quantum stability of155 Landau — Ginzburg model76 Langer modification242 Localised solutions, definition of13 Massive Thirring model213 Massive Thirring model, bound states in strong coupling215—216 Massive Thirring model, equivalence to SG model213—215 Massive Thirring model, O(2) invariance of233 Massive Thirring model, S-matrix of233—235 Massless fermions and chiral symmetry360—361364 Massless fermions and suppression of vacuum tunnelling368 Matrix notation for Yang Mills fields87 Mechanical analogy to non-linear equations17—2024—27 Mechanical analogy to non-linear equations, orbits in24—31 Meron solutions348383384 Monopolessee "'tHooft — Polyakov monopoles" Non-abelian Higgs model59—60352357 Non-linear O(3) model49 Non-linear O(3) model, exact S-matrix of236 Non-linear O(3) model, homotopy analysis in 2 dimensions51—52 Non-linear O(3) model, relevance to ferromagnets5880 Non-linear O(3) model, static solitons in (2 + 1) dimensions57 Non-topological solitons33 Non-topological solitons, static30 Non-topological solitons, time dependent250 Normal modes of fluctuations132 Normal modes of fluctuations in scalar field theory134 Normal modes of fluctuations with zero frequencysee "Zero-modes" Normal modes of fluctuations, around the kink solution140 Normal modes of fluctuations, around the SG soliton201 Normal ordering146—147202 Nuclear democracy209 Operator ordering problem in quantum mechanics243 Operator ordering problem in soliton quantisation258 259 Parity symmetry and axions383 Parity symmetry, violation by -vacua335343382 Particle on a circle (POC)314—317 Particle on a circle (POC), -dependcnt ground state316 Particle on a circle (POC), similarity to -vacua335—336 Path integrals and the energy spectrum170174—175 Path integrals and the stationary phase approximation (SPA)170—171 Path integrals, applied to the oscillator172—174 Path integrals, definition169 Path integrals, Euclidean175—176 Path integrals, gaussian approximation to306 Path-ordered exponentials89357 Periodic potential problem (PPP)303 Periodic potential problem (PPP), derivation of energy band305—311 Periodic potential problem (PPP), instanton of306—307 Periodic potential problem (PPP), properties of304—305 Periodic potential problem (PPP), restoration of symmetry in314 Perturbation scries, Borel summation of375 Perturbation scries, radius of convergence373—379 Perturbation series, divergence of374 Pontryagin index91338341 Pontryagin index and fermion zero modes367 Pontryagin index, current associated with92 Pontryagin index, density of92 Pontryagin index, gauge invariance of97 Prasad — Sommerfield limit69 Quantisation of static solitons by canonical operator methods257—263285—286295—297 Quantisation of static solitons in higher dimensions165 Quantisation of static solitons in the presence of Fermi fields166281—286291—298 Quantisation of static solitons, as applied to the -kink137—150 Quantisation of static solitons, as applied to the sine-Gordon soliton201—203 Quantisation of static solitons, general principles133—135 Quantisation of static solitons, mass renormalisation in144—150 Quantisation of static solitons, non-perturbative aspect of124149 Quantisation of static solitons, using functional integrals178 180281—285 Quantisation of time-dependent solitons in field theory194 Quantisation of time-dependent solitons in non-relativistic quantum mechanics183—193 Quantisation of time-dependent solitons in the presence of Fermi fields286—288 Quantisation of time-dependent solitons, applied to sine-Gordon doublets203—208 Quantum chromodynamics (QCD)2—5 Quantum chromodynamics (QCD), definition of358—359 Quantum chromodynamics (QCD), miniature version of362
model 
-vacua and 
SU(2) symmetry 
) 
) 
- theory, anti-kink 
|
|
Î ïðîåêòå