LAUSR

Classical Field Theory is a graduate level introduction to modern classical field theory, including electromagnetism and general relativity and describes various classical methods for fields with negligible quantum effects. The book is an expanded version of a field theory course in São Paulo. The author focuses on different theoretical solutions that require classical field theory methods, aiming to assist graduate students and researchers in easily make usage of classical field theory methods and adjust to this conceptual level before jumping directly into a course of quantum field theory. The book provides a theoretical framework for understanding fields and the methods associated with them. Most modern methods of classical field theory don’t get included into the standard student curriculum, making this book an invaluable asset. The book makes the transition between classical fields and quantum field theory much smoother and safe. The textbook provides a comprehensive review of classical mechanics, from Lagrangean, equations of motion to the Hamiltonian formalism, conservation laws, canonical transformations and the HamiltonJacobi theory. The author focuses on symmetries, groups, Lie algebras, Abelian Lie group invariance, cyclic groups and representations. The author also reviews the kinematics and dynamics of Special Relativity, Lorentz tensors, covariant Lagrangeans and the Lorentz group. The author defines the notion of field and introduces electromagnetism as a field theory, describing Maxwell’s equations and the EulerLagrange equations. The scalar field, its origins and applications are also present, together with their Lagrangeans, models and the equations of motion. Classical integrability, the continuum limit of discrete, lattice and spin systems are also provided. The book introduces us to the classical perturbation theory, polynomial potential and the formal solutions to the equations of motion. The book also covers the representations of the Lorentz group (Poincaré group, universal cover, Wigner method), rotation matrices, the spin-statistics theorem, Maxwell equation, Abelian vector fields, Proca field, energy-momentum tensor, Maxwell duality, electromagnetic waves and the motion of charged particles. The author describes Hofion solution and Hopf map, complex scalar fields and gauging global symmetries. The book introduces the reader also to the Noether theorem and its applications, nonrelativistic and relativistic fluid dynamics, including fluid vortices and knots. The second part of the book focuses on Solitons and topology as well as Non-Abelian theory (Kink solutions, Sine-Gordon, domain walls and topology, the Skyrmion scalar field, topological numbers, field theory solitons for condensed matter, XY and Rotor model, Spins, superconductivity, Landau-Ginzburg model and the Kosterlitz-Thouless phase transition. The radiation of classical scalar fields (Heisenberg model), Derrick’s theorem, symmetry breaking and the Abelian-Higgs system are also presented, together with the Non-Abelian gauge theory, Yang-Mills equation, Dirac monopole, Diract quantisation, the instanton, nontopological solutions, unstable solitons and moduli space. The last part of the book focuses on other spins or statistics, together with General relativity, from Chern-Simons theory, emergent gauge fields, Quantum Hall effect, Anyon statistics, massive theory, particle-vortex and particle-sting dualities, fermions andDirac spinors or the Dirac equation. The General relativity part contains concepts such as Einstein action and equation, perturbative and non-perturbative gravity, gravitational waves, time-dependent gravity and cosmology, gauges, dimensional reduction, cosmic strings, gravitational instantons and black hole solutions. Each chapter ends with further reading notes and several exercises, allowing the reader to identify the key aspects of the chapter and test their understanding.