We study the moduli space of solitons, scattering of them in the moduli space approximation, and collective coordinate quantization. After a general analysis of the scattering in the moduli space approximation, we consider the example of scattering of two ANO vortices in the Abelian–Higgs model, and find the moduli space metric and interaction potential for the vortices. We then define collective coordinates and, after showing how to change coordinates in a quantum Hamiltonian, we apply to collective coordinates to define their quantization procedure.

We define scalar field theory, explaining its various applications in physics as either fundamental or effective field. We write a general canonical Lagrangian with a potential, and the resulting Klein–Gordon equation.Examples for this are shown, in particular the sine–Gordon model and the Higgs model. For several scalars, we show the O(N) model and, for models with a noncanonical kinetic term, the DBI model and nonlinear sigma models.

We study the Noether theorem, which relates a conserved charge to any global symmetry, and its applications. We define the Noether current and charge, and the general form of the Noether procedure for “gauging” the global symmetry, as well as the ambiguity of the Noether current. We apply it to translations, the Abelian U(1) symmetry, the O(N) model, and vectors in the fundamental of O(N). We define the Noether charge as an integral in terms of fields, and the transformation of the fields as the Poisson bracket of the charge with the fields.

We start by defining the notion of Riemann tensor and curvature, and positive and negative curvature spaces. We then show how to turn a special relativistic invariant theory into a general relativistic invariant one and write down the Einstein–Hilbert action for gravity, based on Einstein's principles and on matching with experiment. We then derive its equations of motion, Einstein's equations. We give examples of usual energy–momentum tensors in curved space and end by interpreting the Einstein's equations.

The BPST-'t Hooft instanton solution is found and explained, as a solution of Euclidean Yang–Mills theory. After setting up the theory, we propose the self-duality equation and show that it minimizes the Euclidean action. On the self-dual condition, the action becomes the second Chern number, the integral of the Chern form and a topological number identified with the instanton number, and a configuration carrying it interpolates between different winding numbers for monopoles. The explicit instanton solution is found by an ansatz, and its action is calculated. We comment on the interpretation in the quantum theory, as governing transitions between different monopole number sectors.

In this chapter, we study the Hopfion solution of electromagnetism. It is a solution characterized by a topological number, the Hopf index, associated with the Hopf map from the 3-sphere to the 2-sphere (Hopf fibration). It is also characterized by nonzero “helicities,” which are defined as integrals of Chern–Simons forms on the spatial volume, as well as having a knot structure, where the electric and magnetic fields are linked, with nonzero linking number. Generalization of these electromagnetic knots are also given.

Here we review special relativity. We define the Lorentz group and Lorentz transformations; then the kinematics of special relativity, defining arbitrary tensors and various common ones; then the relativistic Lorentz force law, for the kinematics of special relativity. We define a relativistically covariant Lagrangian and invariant action for a particle and extend that to a coupling between a current and an electromagnetic gauge field.

The nonpeturbative Schwarzschild solution of the Einstein's equations in vacuum is found. We start with the Newtonian limit for gravity, and find the metric in terms of the Newton potential. We write an ansatz, and solve the equations of motion to find the solution. The Schwarzschild black hole and its event horizon are defined. Birkhoff's theorem for the uniqueness of the Schwarzschild solution is stated.

We define the only example of fully linear general relativity, for plane parallel (pp waves), where the Einstein equation reduces to a Poisson equation. We show, according to Penrose, that in the “Penrose limit,” when focusing in near a null geodesic, we find a pp wave. We show how to go between the “Rosen coordinates” and “Brinkmann coordinates” for the pp wave and give an example of the Penrose limit. Gravitational shockwaves are defined in flat space, leading to the Aichelburg–Sexl metric. General shock waves, shock waves generated by a graviton, and shockwaves in other backgrounds, are defined. We end with the Khan–Penrose interacting solution for two colliding shock waves.

In this chapter, we study the motion of charges and electromagnetic waves. After studying static charges, uniformly moving charges, and the standard electrostatic method of the mirror image charges, we consider the multipole expansion of the electric and magnetic fields. The electric field is generated by monopole (electric charge) and higher multipole, and magnetic field by dipole and higher multipoles. Electromagnetic waves are then studied. For arbitrary moving charges, we calculate the retarded potentials, and in particular the Lienard–Wiechert forms. We then show that we need at least dipoles to generate electromagnetic waves. We end by describing Maxwell duality.

We consider the classical perturbation theory for the equations of motion of a field theory Lagrangian. We consider a scalar field with canonical kinetic term and a potential that contains interactions, and we describe the general formalism. In the case of a polynomial potential, we describe the formal solution and how we can self-consistently solve it in perturbation theory, considering that the potential interaction is small. We construct a diagrammatic procedure for solving it iteratively – that is, the classical limit of the Feynman diagram procedure in quantum field theory, but here it is just a mathematical trick.

In this chapter we consider the examples of the simplest and most common non-Abelian groups, the rotation group SO(3) and the group SU(2). After characterizing them and their representations, we show the equivalence of the two groups in Lie algebra, and the fact that SU(2) is a double cover of SO(3). We also present invariant Lagrangians for the two groups.

We consider radiation from a classical scalar field. Polynomial potentials and the DBI scalar model are analyzed. A source moving at ultra-relativistic speeds (v very close to c) gives a shock wave solution. The solutions for the free scalar, interacting scalar and DBI scalar are found. We sketch the Heisenberg model for the collision of two ultrarelativistic hadrons leading to scalar radiation. The field and radiated energy are calculated.

We define the notion of field, based on the example of electromagnetism. We write the relativistically covariant form of the Maxwell's equations in terms of a gauge field and field strength for it. We define the Euler–Lagrange equations for a field, and based on it, we derive the relativistic Maxwell's equations from a relativistically invariant Maxwell action.

We prove Derrick's theorem about scalar field solitons, then we derive the Bogomolnyi bound for the energy of scalar field configurations in 1+1 dimensions and consider the example of the Higgs system and its kink soliton. Then we consider spontaneous symmetry breaking in the Abelian–Higgs syste, and the different fluctuations and their masses, as well as the description of this system in an unitary gauge. We end with a quick treatment of the non-Abelian Higgs system.

In this chapter, we consider the vortex of the Abelian–Higgs system, the Nielsen–Olesen vortex (or ANO vortex). We find the Bogomolnyi bound for the energy of the system in terms of a topological charge. For a certain relation between coupling, known as a BPS limit, we find that the bound is saturated by a configuration with topological charge, i.e., magnetic charge. In this limit, we find BPS equations, which are solved by a vortex ansatz, for a vortex solution.The properties of the solutionand its application to superconductivity are explored.

We define Chern–Simons gauge fields in 2+1 dimensions, and the quantization of their “level” k. We show that the CS action in a material define a topological response, and gives an integer quantum Hall effect. We show how a CS field emerges, as a statistical field, in materials. We define anyons and show how they can appear from a CS field in the fractional Quantum Hall effect.

In this chapter we study the energy-momentum tensor. After defining it from the Lagrangian formalism, we consider conservation equations in general, and apply it to the energy–momentum tensor. We find an ambiguity in the definition of the energy–momentum tensor, we fix it by considering the symmetric tensor, and we find the interpretation of the tensor's components. The Belinfante tensor form is defined by coupling to gravity. Finally, we give as an example the electromagnetic field, for which we calculate the energy–momentum tensor.