We describe electromagnetism as an Abelian gauge theory (for the Abelian group U(1)). Then we describe it in the language of p-forms, after defining p-forms and their formalism in detail. General p-form fields in various dimensions are also defined. Finally, we consider the massive vector field, the Proca field.

The Friedmann-Lemaître-Robinson-Walker (FLRW) cosmological solution for the expanding (time dependent) Universe is found. We start with an ansatz for a homogenous and isotropic space in comoving coordinates, and define various coordinate systems and analyze the geometry. The Einstein equations reduce to the Friedmann equation for the “Hubble constant” and the acceleration equation for the scale factor, related through the conservation of the energy-momentum tensor. Given an equation of state for matter, we can solve the Friedmann equation.

We consider field theory solitons relevant for condensed matter. We start with a field theory arising from a two-dimensional system of spins, the XY model, leading to the “rotor model,” or “O(2) model”. From the bosonic Hubbard model, we show a representation that leads to the same quantum rotor model. In the continuum limit, we obtain a massless scalar that has a global vortex as its solution. The dynamics of these vortices is relevant for the Kosterlitz–Thouless (KT) phase transition, a quantum phase transition appearing for instance in 2+1 dimensional superconductivity. The bosonic Hubbard model leads, in the continuum limit, also to a relativistic Landau–Ginzburg model, that has a kink-like solution.

We treat “classical spinors”, in the sense of studying the classical equations of motion, though all fermions are quantum. We start by defining Weyl and Dirac spinors in various ways. We then define gamma matrices, satisfying the Clifford algebra, and prove various gamma matrix identities. We finally define Majorana spinors.

We find the Lie algebra of the Lorentz group and then extend it to the Poincaré group, the group of symmetries of flat space. We then point out that, as SU(2) is the universal cover of SO(3), for the Lorentz group SO(3,1) the universal cover is SL(2,C).We then use Wigner's method, using the little group in four dimensions, to find massive and massless representations of the Lorentz and Poincaré groups. We thus find various possible fields, corresponding to these representations. We end by explaining how SL(2,C) is the universal cover of SO(3,1).

We study the 't Hooft-Polyakov monopole solution of the nonabelian Georgi-Glashow model, a model with the gauge group SU(2)=SO(3) and scalar fields in the 3 representation. After setting up the model, and finding the vacuum manifold, we solve for the monopole through an ansatz. We then study the topology of the solution through an analysis of homotopy groups. We derive a Bogomolnyi bound, and a BPS limit (for the scalar quartic coupling to vanish), in which we get linear BPS equations, which are solved exactly. The topology of the BPS monopole is compared with the topology of the Dirac monopole, and an embedding relation of the latter in the former is found.

We describe vector theories in 2+1 dimensions, Maxwell, Proca, with topological mass terms, and finally the “self-dual” action. We generalize the “self-duality” to odd dimensions. We show the duality of the self-dual action to the one with topological mass term, through the intermediary of a master action. We generalize the proof to odd dimensions. We then describe the basic 1+1 dimensional example of duality, the “T-duality”, and finish with comments on CS fields in higher odd dimensions.

In the first chapter, the most important concepts of classical mechanics are quickly reviewed. The Lagrangian and Hamiltonian formalism are described. The way to deal with systems with constraints is described. Poisson brackets and the use of canonical transformations in the Hamiltonian formalism, as well as the basics of Hamilton–Jacobi theory complete this chapter.

We present examples of nonrelativistic field theories, starting with the nonrelativistic limit of a scalar field with canonical kinetic term. Then we present hydrodynamics, the study of fluids, with the goal of describing water waves. We derive the KdV equation and its soliton from the description of water waves. The KS equation is also described. Finally, we describe surface growth and the KPZ equation.

We define the Dirac monopole as a simple consequence of extending Maxwell duality to the Maxwell equations with sources, and we show that the resulting gauge fields are only defined on patches. We write formulas in terms of p-form language, and define the magnetic charge in terms of the gauge fields on patches. Then, from the quantization of the first Chern number, a topological number, we obtain Dirac quantization for the product of electric and magnetic charges. One obtains an unphysical Dirac string singularity, and its unphysical nature leads again to Dirac quantization. Finally, semiclassical nonrelativistic considerations also lead to the same Dirac quantization.

We define nonabelian gauge theory. We start by defining nonabelian gauge groups and their properties, then (minimal) coupling other fields to the gauge field, through a covariant derivative. The action and gauge invariance of the pure Yang-Mills theory is given, and the resulting Yang-Mills equation (equation of motion) is derived.

We start by defining the vielbein-spin connection formulation of general relativity and the Palatini formalism. Next we define the Taub–NUT solutions and their analytical continuation, the Euclidean gravitational instanton defined by Hawking. Next, following the example of the Yang–Mills instanton, we write the Einstein equations in Euclidean signature as self-duality equations for the spin connection, which we solve by an instanton ansatz, obtaining the Eguchi–Hanson metric, and example of ALE space. We rewrite it and generalize it in the form of the Gibbons–Hawking multi-instanton solution.

We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.

We consider “dimensionally reduced” gravitational solutions. We write a domain wall ansatz and solve the Einstein equations for it, first for a perturbative nonrelativistic solution, and then for a nonperturbative relativistic one. We write a cosmic string ansatz and solve Einsein's equations by dimensional reduction to 2+1 dimensions, and alternatively in the weak field limit. We define the cosmological constant and write an ansatz for a 2+1 dimensional black hole in a space with cosmological constant, obtaining the BTZ black hole solution. Anti–de Sitter space is defined in general, starting from the BTZ black hole for M = –1.

We define general relativity. We first consider intrinsically curved spaces and the notion of metric. Einstein's theory of general relativity is defined, based on the two physical assumptions, that gravity is geometry, and that matter sources gravity, and leading to general coordinate invariance and the equivalence principle. Kinematics, specifically tensors, Christoffel symbol and covariant derivatives, is defined. The motion of a free particle in a gravitational field is calculated.