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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
Quantum chromodynamics (QCD), possible confinement in348351 Quantum electrodynamics (QED) of a charged scalar field357 Quantum electrodynamics (QED), divergence of perturbation series374 Quarks in QCD359 Quarks, confinement of351357—358 Quarks, flavours of359 Quarks, massless360 361 Rapidity variables226 Rapidity variables, unitarity, analyticity and crossing in226—227 Renormalisation of quantum kink mass144—150 Renormalisation of SG soliton mass202—203 Renormalisation of the gauge coupling constant346—347 Renormalisation of Yang — Mills instanton contribution345—347 Restoration of symmetry by instantons in PPP314 Restoration of symmetry in the isotropic ferromagnet80 Restoration of symmetry, absence of in SG model317—318 Running gauge-coupling-constant8346—347 S-matrix theory8 S-matrix theory, application in (1 + 1) dimensions225—236see Self-duality condition99 Self-duality condition, analogues in other models5570120 Semi-classical method and QCD4—5302 Semi-classical method and tunnelling306—314 Semi-classical method and zero modes142150—152237—248 Semi-classical method for quantising periodic solitons187—195 Semi-classical method for quantising static solitons137—166 Semi-classical method in the presence of Fermi fields281—302 Semi-classical method, condition for validity196—197 Semi-classical method, general limitations of4196 Semi-classical models of hadrons4 Semi-classical models of hadrons, theoretical framework289—291302 Sine-Gordon (SG) solitons, as fermions199213—219 Sine-Gordon (SG) solitons, quantisation of201—203 Sine-Gordon (SG) solitons, the classical solution37 Sine-Gordon (SG) system35 Sine-Gordon (SG) system, anti-soliton of37 Sine-Gordon (SG) system, classical solutions of37—40 Sine-Gordon (SG) system, doublets or breathers of40 Sine-Gordon (SG) system, equivalence to massive thirring model213—215 Sine-Gordon (SG) system, exact S-matrix of233—235 Sine-Gordon (SG) system, O(2) invariance of233 Sine-Gordon (SG) system, quantisation of201—213 Sine-Gordon (SG) system, soliton of37 Sine-Gordon (SG) system, two-soliton solutions38—40 Size of solitons/instantons in theory22 Size of solitons/instantons in PPP313 Size of solitons/instantons in the model122 Size of solitons/instantons in the abelian Higgs model78114331 Size of solitons/instantons in the O(3) model58 Size of solitons/instantons in the Yang — Mills system105344—345 Size of solitons/instantons, importance of313344—345 Solitary waves in (1 + 1) dimensions16—31 Solitary waves in 2-dimensional isotropic ferromagnets80 Solitary waves of the abelian Higgs model75—78 Solitary waves of the O(3) model57—58 Solitary waves under Lorentz transformations22—23125 Solitary waves, as distinct from solitons15—16 Solitary waves, being often referred to as solitons15—1645 Solitary waves, definition of13—14 Solitary waves, long range forces between81—82 Solitary waves, quantisation ofsee "Quantisation" Solitary waves, static solutions as14 Soliton creation operator218—219 Soliton sector of states153 Soliton sector of states with half-integral charge295—298 Soliton sector of states, canonical commutation rules in162—164 Soliton sector of states, energy levels of kink sector141—142 Soliton sector of states, stability of155—156 Soliton sector of states, topological quantum number of155 Solitonssee "Solitary waves" Solitons in quantum SG theory201—203 Solitons of the SG system37 Solitons, definition of14 Solitons, distinction from solitary waves15 Solitons, meaning in popular usage15—16 Spin from isospin73 Spontaneous symmetry breaking in (1 + 1) dimensional -theory13139 Spontaneous symmetry breaking in the Higgs phenomenon318—319 Spontaneous symmetry breaking in the O(3) model50 Spontaneous symmetry breaking in the sine-Gordon theory200317 Spontaneous symmetry breaking of chiral SU(2) SU(2)360 Stability angles, analogy to Floquet indices287 Stability angles, definition of189—190 Stability angles, stability angles in Held theory194 Stationary phase approximation (SPA) and the Bohr — Sommerfeld condition183—187 Stationary phase approximation (SPA) in field theory194 Stationary phase approximation (SPA), definition170
Stereographic projection56 Time reversal symmetry and axions383 Time reversal symmetry, violation by -vacua343382 Topological index or charge and fermionic zero modes298367 Topological index or charge for scalar fields in (1 + 1) dimensions31—34 Topological index or charge for the monopole solutions64—65 Topological index or charge for the O(3) model51—5254 Topological index or charge for Yang — Mills instantons91 Topological index or charge, as quantum numbers155—156 Topological index or charge, conservation of, in (1 + 1) dimensions32—33 Topological index or charge, currents associated with336492 Topological vacua and cluster decomposition370—372 Topological vacua, definition of325 Topological vacua, suppression of tunnelling362—370 Topological vacua, tunnelling between331—332341—343 Translation mode in instanton calculations307—308390—393 Translation mode in kink quantisation142 Translation mode, origin of150—151 Translation mode, treatment of252—263 Tunnelling and the semi-classical method and real-time amplitudes312384 Tunnelling and the semi-classical method in abelian gauge theory325332 Tunnelling and the semi-classical method in non-relativistic quantum mechanics130303—318 Tunnelling and the semi-classical method in the presence of fermions367—370 Tunnelling and the semi-classical method in Yang — Mills theory340—341 Tunnelling and the semi-classical method, tunnelling barrier314340—341 Turning points of orbits187189388—389 U(1) problem361see Unitarity225 Unitarity for (1 + 1) dimensional O(N) models227 Vacuum angle ""334341 Vacuum angle "", as a parameter in the Lagrangian335343 Vacuum of QCD with massless fermions370—373 Vacuum of the abelian Higgs model320325 Vacuum of the symmetric theory136 Vacuum of the symmetry-broken theory138—139 Vacuum of the Yang — Mills theory341 Vacuum sector of states in the theory138—139 Vacuum sector of states in the sine-Gordon theory200 Vacuum sector of states, doublet states in208 Vacuum tunelling in abelian Higgs model331—332 Vacuum tunelling in higher SU(N) groups343 Vacuum tunelling in Yang — Mills theory341—343 Vacuum tunelling, suppression by massless fermions362—370 Virial theorem47—48 Virial theorem and the Euclidean SG model318 Virial theorem and the O(3) model48—49 Weak coupling condition and the Yang — Mills system347—350 Weak coupling condition in non-relativistic quantum mechanics128 Weak coupling condition in scalar field theory134 Weak coupling condition, relationship to small-h condition197—198 Wilson loop, abelian case353381 Wilson loop, non-abelian case357381 WKB methodsee "Semi-classical method" WKB Method and the Bohr — Sommerfeld condition186 WKB Method and weak coupling196—198 WKB Method in field theory187—195 WKB Method, accuracy of, for anharmonic oscillator196 WKB Method, exactness of results for the SG bound states211—213 WKB Method, fluctuation factor385—389 WKB Method, using path integrals183—187 Yang — Mills theory, -vacua and sectors in341—343 Yang — Mills theory, definition87 Yang — Mills theory, gauge equivalence in340 Yang — Mills theory, gauge transformations in87 Yang — Mills theory, Gauss' theorem in339 Yang — Mills theory, homotopy classification89—98338 Yang — Mills theory, instantons of102—111 Yang — Mills theory, large instanton problem in344—347 Yang — Mills theory, running coupling constant in346—347 Yang — Mills theory, topological vacua of340 Yang — Mills theory, vacuum tunnelling in341—343 Zero energy fermion modes and suppression of tunnelling367—368 Zero energy fermion modes in the presence of single instanton368 Zero energy fermion modes, consequences of, in soliton quantisation295—298 Zero energy fermion modes, example in (1 + 1) dimensions293—294 Zero energy fermion modes, examples in higher dimensions298367—368 Zero energy fermion modes, relation to Pontryagin index367 Zero modes in boson field theory and kink quantisation142150—153 Zero modes in boson field theory in a U(1) symmetric model244 Zero modes in boson field theory in instanton physics345381393 Zero modes in boson field theory, due to translation symmetry254 Zero modes in boson field theory, principles behind treatment of237—243 Zero modes in boson field theory, qualitative discussion150—153 Zero modes in boson field theory, treatment in field theory252—263 Zero modes in non-relativistic quantum mechanics in 1-dimension131 Zero modes in non-relativistic quantum mechanics in 2-dimensions237—243 Zero modes in non-relativistic quantum mechanics in instanton calculations307—308390
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