Solitons in Field Theory and Nonlinear Analysis

by Yang, Yisong

• ISBN13: 9780387952420
• ISBN10: 038795242X
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: Springer Verlag

Summary

This book is on soliton solutions of elliptical partial differential equations arising in quantum field theory, such as vortices, instantons, monopoles, dyons, and cosmic strings. The book presents in-depth description of the problems of current interest, forging a link between mathematical analysis and physics and seeking to stimulate further research in the area. Physically, it touches the major branches of field theory: classical mechanics, special relativity, Maxwell equations, superconductivity, Yang-Mills gauge theory, general relativity, and cosmology. Mathematically, it involves Riemannian geometry, Lie groups and Lie algebras, algebraic topology (characteristic classes and homotropy) and emphasizes modern nonlinear functional analysis. There are many interesting and challenging problems in the area of classical field theory, and while this area has long been of interest to algebraists, geometers, and topologists, it has gradually begun to attract the attention of more analysts. This book written for researchers and graduate students will appeal to high-energy and condensed-matter physicists, mathematicians, and mathematical scientists.

Table of Contents

Preface	vii
Notation and Convention	xxiii
Primer of Field Theory	1	(42)
Mechanics and Fields	1	(7)
Action principle in classical mechanics	2	(3)
Charged particle in electromagnetic field	5	(1)
Schrodinger equation via first quantization	6	(2)
Relativistic Dynamics and Electromagnetism	8	(7)
Minkowski spacetime and relativistic mechanics	8	(4)
Klein-Gordon fields	12	(1)
Maxwell equations	12	(3)
Scalar Fields and Symmetry	15	(5)
Variational formalism	15	(1)
Noether's theorem and conserved quantities	16	(3)
Static solutions and Derrick's theorem	19	(1)
Gauge Field Theory	20	(7)
Local symmetry and gauge fields	20	(4)
Low temperature and spontaneous symmetry-breaking	24	(1)
Goldstone particles and Higgs Mechanism	25	(2)
Yang-Mills Fields	27	(3)
General Relativity and Cosmology	30	(11)
Einstein field equations	30	(7)
Cosmological consequences	37	(4)
Remarks	41	(2)
Sigma Models	43	(36)
Sigma Model and Belavin-Polyakov Solution	43	(7)
Sigma model for Heisenberg ferromagnet	43	(3)
Solution by rational functions	46	(2)
Topology	48	(2)
Gauged Sigma Model	50	(6)
Field theory and self-dual equations	50	(3)
Multisolitions: existence theorems	53	(3)
Governing Equations and Characterization	56	(1)
Mathematical Analysis	57	(19)
Regularized equation and range of parameter	58	(1)
Subsolution and variational methods	59	(8)
Existence of supersolution	67	(1)
Existence of bounded solution	68	(1)
Asymptotic limit	69	(2)
Recovery of original field configurations	71	(1)
Magnetic flux and minimum energy value	71	(1)
Brouwer degree of map	71	(3)
Nonexistence of solution of unit degree	74	(2)
Remarks	76	(3)
Multiple Instantons and Characteristic Classes	79	(42)
Classical Yang-Mills Fields	79	(9)
Action principle and self-dual equations	80	(3)
Energetic and topological characterizations	83	(2)
't Hooft instantons	85	(3)
Liouville Equation and Solution	88	(4)
Liouville method	88	(2)
Backlund transformation method	90	(2)
Witten's Instanton	92	(3)
Field configurations and equations	92	(2)
Explicit instanton solutions	94	(1)
Instantons and Characteristic Classes	95	(8)
Self-duality and Witten--Tchrakian equations	95	(7)
Quasilinear elliptic equation	102	(1)
Existence of Weak Solution	103	(4)
Asymptotic Estimates	107	(9)
Topological Charge	116	(1)
Remarks	117	(4)
Generalized Abelian Higgs Equations	121	(36)
Field Theory Structure	121	(6)
Formulation and main existence theorem	122	(3)
Nonlinear elliptic system	125	(2)
General Problems and Solutions	127	(3)
Compact Surface Case	130	(5)
Necessary condition	130	(1)
Variational principle	130	(3)
Existence of solution	133	(1)
Uniqueness	134	(1)
Solution on Plane: Existence	135	(5)
Variational problem	135	(1)
Coercivity	136	(3)
Existence and uniqueness of critical point	139	(1)
Solution on Plane: Asymptotic Behavior	140	(4)
Pointwise decay near infinity	141	(1)
Exponential decay estimates	142	(1)
Uniqueness and quantized integrals	143	(1)
Nonexistence Results	144	(7)
Arbitrary Coefficient Matrix Case	151	(4)
Remarks	155	(2)
Chern-Simons Systems: Abelian Case	157	(54)
Schrodinger Equation	157	(7)
Schrodinger fields and Chern-Simons dynamics	158	(2)
Explicit static solution	160	(4)
Relativistic Chern-Simons Model on Plane	164	(3)
Field equations and existence results	164	(2)
Topological lower energy bound	166	(1)
Construction of Solution	167	(10)
Iterative method and control of sequence	168	(5)
Global convergence theorems	173	(4)
Symmetric Non-topological Solutions	177	(9)
Existence theorem	178	(1)
Two-point boundry value problem	179	(1)
Shooting analysis	180	(6)
Solutions on Doubly Periodic Domains	186	(14)
Boundary condition modulo gauge symmetry	186	(2)
Existence versus coupling parameter	188	(1)
Construction via sub- and supersolutions	189	(5)
Alternative variational treatment	194	(6)
Tarantello's Secondary Solution	200	(8)
Critical coupling parameter	200	(2)
Local minimum	202	(3)
Nonminimum via mountain-pass lemma	205	(3)
Remarks	208	(3)
Chern-Simons Systems: Non-Abelian Case	211	(42)
Lie Algebras and Cartan-Weyl Bases	211	(10)
Simple examples	212	(2)
Classification theorem	214	(5)
Root vectors and Cartan matrices	219	(2)
Non-Abelian Gauged Schrodinger Equations	221	(11)
Adjoint representation and elliptic problems	221	(5)
Toda systems	226	(5)
Explicit non-Abelian solutions	231	(1)
Relativistic Chern-Simons Systems	232	(4)
Elliptic System and its Variational Principle	236	(5)
Existence of Minimizer	241	(8)
Boundary condition	241	(1)
Minimization	242	(3)
Asymptotic behavior	245	(3)
Quantized integrals	248	(1)
Original field configuration	248	(1)
Some Examples	249	(2)
Remarks	251	(2)
Electroweak Vortices	253	(46)
Massive non-Abelian Gauge Theory	253	(10)
Governing Equations	253	(3)
Periodic boundary condition	256	(2)
First-order system and existence theorem	258	(2)
Variational proof	260	(3)
Classical Electroweak Theory	263	(6)
Unitary gauge framework	263	(2)
't Hooft periodic boundary conditions	265	(3)
Lower energy bound and its saturation	268	(1)
Multi-constrained Variational Approach	269	(8)
Elliptic equations	269	(1)
Existence via minimization	270	(4)
Alternative formulation	274	(3)
Two-Higgs Model	277	(19)
Physical background	277	(1)
Field theory model and equations	277	(2)
Periodic multivortices	279	(7)
Planar solutions	286	(10)
Remarks	296	(3)
Dyons	299	(54)
Dirac Monopole	299	(6)
Electromagnetic duality	300	(1)
Dirac strings and charge quantization	301	(2)
Fiber bundle device and removal of strings	303	(2)
Schwinger Theory	305	(2)
Rotation symmetry	305	(1)
Charge quantization formula for dyons	305	(2)
Julia--Zee Dyons	307	(15)
Field equations	307	(2)
Explicit solutions in BPS limit	309	(2)
Existence result in general	311	(11)
Weinberg-Salam Electroweak Dyons	322	(3)
Radial Equations and Action Principle	325	(1)
Constrained Variational Method	326	(24)
Admissible space	326	(4)
Partial coerciveness and minimization	330	(8)
Weak solutions of governing equations	338	(3)
Full set of boundary conditions	341	(3)
Asymptotic estimates	344	(4)
Electric and magnetic charges	348	(2)
Remarks	350	(3)
Ordinary Differential Equations	353	(18)
Existence Results	353	(2)
Dynamical Analysis	355	(12)
Local solution via contractive mapping	355	(3)
Parameter sets	358	(4)
Asympotic limits	362	(2)
Continuous dependence	364	(1)
Critical behavior and conclusion of proof	364	(3)
Applications	367	(2)
Remarks	369	(2)
Strings in Cosmology	371	(68)
Strings, Concial Geometry, and Deficit Angle	371	(7)
Localized energy distribution and multiple strings	372	(2)
Harmonic map model	374	(4)
Strings and Abelian Gauge Fields	378	(9)
Governing equations over Riemann surfaces	378	(2)
Role of defects	380	(2)
Obstructions to existence	382	(1)
Proof of equivalence and consequences	383	(4)
Existence of Strings: Compact Case	387	(8)
Existence for N ≥ 3	387	(7)
Existence for N = 2 and nonexistence for N = 1	394	(1)
Existence of Strings: Noncompact Case	395	(14)
Existence results	395	(1)
Construction of solutions	396	(7)
Asymptotic decay estimates	403	(6)
Symmetric Solutions	409	(7)
Necessary and sufficient condition for existence	409	(1)
Equivalence theorem	410	(1)
N-strings	411	(5)
Symmetric Solution on S2	416	(9)
Balanced strings at opposite poles	416	(1)
Differential equation	417	(1)
Solution on P	418	(4)
Solutions on full S2	422	(1)
Nonexistence of unbalanced solutions	422	(3)
Non-Abelian Cosmic Strings	425	(11)
Massive W-boson and strings	426	(3)
Einstein--Weinberg--Salam system	429	(7)
Remarks	436	(3)
Vortices and Antivortices	439	(44)
Gauge Field Theory and Coexisting Strings	439	(6)
Action density	440	(3)
Existence theorems	443	(2)
Simplification of Equations	445	(5)
Proof of Existence	450	(15)
Vortices and antivortices	450	(3)
Strings and antistrings	453	(9)
Asymptotic estimates	462	(3)
Quantized Flux, Total Curvature, and Topology	465	(4)
Unique Solutions on Compact Surfaces	469	(12)
Formulation on line bundles	470	(2)
Number count	472	(1)
Solution and fixed-point method	473	(8)
Remarks	481	(2)
Born--Infeld Solutions	483	(42)
Nonlinear Electromagnetism	483	(10)
Point charge problem	484	(4)
Bernstein theorms	488	(5)
Relation of Electrostatic and Magnetostatic Fields	493	(8)
Electrostatic fields	493	(1)
Magnetostatic fields	494	(2)
Generalized Bernstein problem	496	(4)
Mixed interaction case	500	(1)
Nonlinear Wave Equations	501	(7)
Static solutions	501	(4)
In view of Nambu--Goto string theory	505	(3)
Abelian Strings	508	(14)
Existence and uniqueness theorems	508	(8)
Analysis of compact surface case	516	(3)
Solutions on noncompact surfaces	519	(3)
Remarks	522	(3)
References	525	(24)
Index	549