This is a summary of the courses PHY 610-611 as taught with the text Fields. It is meant to give the general ideas of the course as motivation. All this will be repeated later in the course/text with detailed explanations & examples, but is given here for emphasis, & to avoid it getting lost among the trees. See also the Outline in Fields for a longer but more telegraphic list, & the Preface there for what's unique about this text.

The course

This is a second-year graduate course. This means: In spite of what you may have heard, getting an exposure to quantum field theory before taking this course will not help. (However, an exposure to particle physics, @ least @ an undergraduate level, is almost a prerequisite.) You can waste your time looking for a placebo, learning legacy topics that everybody else teaches but which are useless in real-life applications. But again, once you get to research, you'll have to learn how to tackle new ideas, & that conformism will not help you make your mark. For example: (See the beginning of my Fields page, or the Preface of that book, for more details.) These are not "math" or "tricks", these are standard tools for LHC physics. Don't let someone trick you into thinking that 1970's baggage is enough to understand modern physics. Calculations are important for comparing to experiment, but calculational methods also give insight into new physical principles.

The general subject matter

Field theory is the description of particle physics in a way that calculations can be done & compared with experiment. In particular this means the Standard Model, but also some proposed extensions. For example, Quantum Electrodynamics is the most accurate theory of nature known, making successful predictions to over 12 decimal places. The most accurate calculations have been done with perturbation theory, using classical field theory as the lowest order in this perturbation expansion.

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.

Part One: Symmetry

Special relativity is a crucial part of relativistic quantum field theory. It is not about just Lorentz transformations. It is about representations of the Poincaré group, & how to conveniently deal with them. We give a basic discussion of the group theory useful in dealing with it, & with the "internal symmetry" used to classify & relate different kinds of particles. A useful generalization of the Poincaré group, & an approximate symmetry of nature, is the conformal group, which is actually simpler in some respects, & often more useful because of its stronger constraints.

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.

Part Two: Quanta

The most versatile method to quantize field theory is path (functional) integrals: It's more rigorous, easier, manifestly Lorentz covariant, more general, & more pedagogical than canonical quantization. It's the one everybody uses, even though it isn't the one everyone teaches. (An interesting analogy: 3D physics is usually taught in Gibbs' notation, but 4D physics is always taught with index notation, which works for all dimensions.) We use this method to find S-matrices, & thus cross sections, for general field theories. We use Feynman diagrams (graphs) to describe the general features, such as unitarity & causality. The group theory of internal symmetry can also be calculated diagrammatically.

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.

Higher Spin

(Stop here if you're only interested in the field theory course.)

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.