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.

We consider perturbative gravity for small fluctuations of the metric and derive the quadratic Fierz–Pauli action for them. We derive its gauge invariance and fix it by the de Donder gauge, equivalent of the Lorenz gauge of electromagnetism, and leading to the KG equation. We also define synchronous and Newtonian gauges. Gravitational waves are defined, and their generation via retarded sources is shown. The formula for the gravitational wave perturbation in terms of the time variation of a quadrupole momentum is proven. Finally, the exact cylindrical wave of Einstein and Rosen is found.

We calculate “deflection of light by the Sun." First, we define a first-order action for a massless particle moving in a gravitational field, and then we calculate the motion of light on a geodesic as motion of light in a medium with a small, position-dependent index of refraction, giving the light deviation for small angles. Then we redo the calculation from the Hamilton–Jacobi formalism by first defining the Hamilton–Jacobi equation for light motion and then solving it. This gives the nonperturbative light deviation that matches the previous calculation at small angles. We end by comparison with the deflection of light by the Sun in special relativity, which is different by a factor of 2.

We consider first Lorentz invariant Lagrangians for spinors, Dirac, Majorana, and Weyl. The Dirac Lagrangian leads to the Dirac equation, whose solutions we study.We find the solutions u(p) and v(p) and calculate their normalizations.

We study dualities in various dimensions. We start with the example of Maxwell duality in 3+1 dimensions from the point of view of a transformation with a master action. We continue with the particle–vortex duality in 2+1 dimensions, also through a master action, and then the particle–string duality in 3+1 dimensions. Finally, we describe the general Poincaré duality in d dimensions, and we end with applications in 3+1 dimensions.

We study complex scalar fields and their couplings. A complex scalar with a global U(1) invariance has an electric current and associated charge, and we can “gauge” this symmetry, i.e., make it local. The procedure for making it local is the Noether procedure, and it amounts to making derivatives covariant with respect to a gauge field (minimal coupling to the gauge field), plus adding more terms.

We consider “kink” solutions of 1+1 dimensional scalar field models with a canonical kinetic term and a potential for the scalar. Classical, static, soliton solutions are found as solutions of the classical mechanics motion in an inverted (V = -U) potential. The cases of the $\phi^4$ and Higgs models are analyzed, and the Higgs “kink” solution is found and described. The sine–Gordon model is also analyzed and its soliton found. The topology of the solutions is characterized. Finally, embedding of the solution in higher dimensions as a “domain wall” is also considered.

We consider fluid dynamics and solutions. We define the ideal fluid and viscous fluid dynamics (governed by the Navier–Stokes equations) and their relativistic generalizations. The notion of vorticity and fluid helicity is defined, and the wave of small fluid fluctuations is found. Finally, we define fluid vortices and knotted solutions.

A general topological classification accross theories and dimensions is described. First for scalars, then for gauge fields, in terms of various homotopy groups, we classify all the solitons described.A generalization of Derrick's argument in the presence of gauge fields is given. We then show, with examples, how to reduce the theory in the presence of a specific ansatz for a solution.

We describe nontopological solitons, specifically a Q-ball, which is a scalar field soliton with a nonzero conserved (global or local) charge Q. We find the condition to find a Q-ball and its equation of motion. Next we study sphalerons, unstable solitons that interpolate between two vacua in the space of field configurations by passing through a saddle point. We give the example of a real Higgs model with a circle domain, and find the explicit “sphaleron on a circle.” Finally, we comment on the complex kink as a sphaleron, and on sphalerons in electroweak theory.

We consider the Skyrmion solution of classical field theory. We define the Skyrme model as the extension of the nonlinear sigma model of QCD by the addition of a new “Skyrme term”. We analyze the model and define a topological “winding number” for the scalars in spacetime. The Skyrmion solution is found by imposing a “hedgehog” ansatz for the scalars. Generalizations of the model are studied, the Skyrme–Faddeev model and the DBI–Skyrme model, for which we identify the solution, the DBI–Skyrmion.

We examine Poisson brackets in field theory and the symplectic formulation of Hamiltonian dynamics. We start by describing the symplectic formulation of classical mechanics. Then we generalize it and Poisson brackets to field theory. As examples of the formalism, we consider a scalar field with canonical kinetic term and the nonlinear sigma model.

We describe the notion of integrability in classical physics and the possible existence of a Lax pair. As examples of integrable systems, we describe Toda and Calogero–Moser systems and Toda field theory. Field theories are also obtained from discrete systems – for instance, spin systems. The Heisenberg model is described, and the resulting scalar field model obtained.