Fields is divided into 3 topics: Symmetry (PHY 610), Quanta (PHY 611), & Higher Spin (parts of which may appear in other courses). "Symmetry" is mostly about relativistic classical field theory, but also relativistic classical mechanics, & includes a minimal amount of group theory to understand these topics & particle physics. "Quanta" is about actual calculations in quantum field theory, and the results; most is in the form of S-matrix elements, & the resultant cross sections measured by detectors in particle accelerators (& elsewhere). "Higher Spin" discusses general relativity, strings, & other topics that do more than simply extend the Standard Model.
In particular, we describe 2-component spinor notation in detail. "Notation" is not just language, but how you make symmetry clear, & algebra simple. We give the simplest version of this notation, using only 2-component spinor indices, thus avoiding 4-vector indices, the useless "σ-matrices", & their pointless Fierz identities. In particular, we'll find this the easiest way to discuss supersymmetry, an extension of the Poincaré group that mixes fermions & bosons, but later we'll see that even calculations involving only vector fields are practical only in this notation. Another application is a simple relativistic wave equation that describes all spins (with, e.g., the Dirac & Maxwell equations as special cases).
The basic forces of nature are described by theories with ("gauge") symmetries that act independently ("locally") @ each point in spacetime. We discuss such theories & some of their solutions. As an introductory example, we cover the relativistic point particle in external fields: its classical mechanics is analogous to a gauge theory in a one-dimensional space-time (the "worldline"). Electron-positron pair creation can be described by this simple approach.
We then put all these things together to discuss the Standard Model & its extensions. One important case is the Higgs mechanism, which describes how particles get mass. This is closely related to the concept of breaking of chiral symmetry, an approximate symmetry useful @ energies below the confinement scale but above the masses of the lightest quarks. Grand Unified Theories propose a simpler treatment of the fermions and forces. We then construct supersymmetric theories by the use of superspace, which not only allows a simple description of classical supersymmetric theories but will also be used later to make quantum calculations simpler than in nonsupersymmetric theories.
Quantization of gauge theories requires some special techniques. There are many different gauges useful for different purposes (hence the whole point of gauge symmetry). A general method for gauge fixing is that of Becchi, Rouet, Stora, & Tyutin, which we derive for general quantum mechanics, & use to explain the appearance of "ghosts". We then apply these techniques to calculate some tree amplitudes in various gauge theories, including QCD, QED, & some supersymmetric theories. In particular, we apply the spacecone gauge to explicitly find 4 & 5-gluon amplitudes, which are almost trivial using the methods introduced, but would be harder with the older "spinor helicity" methods, and prohibitively difficult with 4-vector notation in Lorenz gauges.
We then generalize to the quantum corrections to these "semiclassical" results by describing "loop" diagrams. Integration over the momenta circulating in these loops is generally divergent @ large ("UV") momenta, so we begin with the general procedure of how to remove these infinities. The most general, easiest, & manifestly symmetric method is dimensional regularization: For all intents & purposes it's the only one used in actual calculations @ more than 1 loop. We then give some 1 & 2-loop examples, & discuss some physical consequences, such as symmetry breaking, energy-dependence of effective coupling constants, & how to make fermions out of bosons in two dimensions. We also discuss the summation of the perturbation expansion, which fails, & can only be fixed if the UV divergences did not appear in the first place.
Important improvements appear upon the introduction of vector fields in the loops, including the possibility of cancellation of UV divergences. GUTs get an experimental verification from the convergence of the couplings of the various forces to a single value @ high energies, especially in the supersymmetric case. Another important effect is the violation of gauge symmetry unless fermions satisfy certain constraints, verified experimentally by the Standard Model. Lattice gauge theory is a nonperturbative approach with some success in describing confinement. The parton model, as predicted by QCD, allows the calculation of the "weak" parts of strongly-interacting processes. The first-quantized methods described previously can also be applied to loops.
To go far beyond the Standard Model, we need to understand spins > 1. In particular, general relativity is the study of spin 2. We consider an approach that allows coupling to all the fields of the Standard Model, not just point particles & electromagnetism. In particular Weyl symmetry, the local generalization of conformal symmetry, is the simplest way to construct solutions applicable to astronomy & cosmology.
Supergravity is the supersymmetric generalization of gravity, & the only nontrivial way to describe spin 3/2. We give descriptions here, with & without superspace. The Higgs mechanism for supersymmetry gives mass to spin 3/2. A closely related topic is the derivation of theories from higher-dimensional ones, since many supersymmetric theories become simpler in some ways in higher dimensions.
String theory is an active area of research, attempting to generalize point particles to objects with one-dimensional extent in space. Most of it is speculative, so we give a limited approach (but including loops). In particular, the only experimental evidence of strings comes from the scattering & spectrum of hadrons, which are related by Regge theory. The naive string theories fail in the description of high-energy fixed-angle scattering, whose origin we explain by examining the parton model implied by the string action.
String theories provide specific interacting models of higher spin. But free theories of higher spin can be described in full generality. The equations of motion were described earlier; here we give the gauge-invariant actions. Again a simple expression can be applied to arbitrary representations of the Poincaré group in arbitrary dimensions. (String field theory is a special case.) This method is based on BRST, & hence also automatically gives the gauge fixing, with all necessary ghosts.