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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.
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 “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.
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.
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.
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 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 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.
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.
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 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 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.
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.
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 solution and its application to superconductivity are explored.
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.
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.