Preparation for research

There is a gap between standard courses and what you need to know to begin research in high energy theory. This is a list of some of these topics, with a few suggested references. "Theory" is here meant in the narrowest sense, excluding both phenomenology and model building, and nowadays means mostly string theory and related topics.

Topics

General Relativity

Einstein's theory of gravity is important not only in its own right, but for applications to astrophysics, cosmology, string theory, and even model building in particle physics. Although it is not necessary to take a semester graduate couse in the topic, at least the basics are assumed in many research areas (and even some advanced courses):
  1. Local symmetries: coordinate transformations, local Lorentz transformations, local Weyl scale transformations.
  2. Vierbeins (tetrads, local frames): The vierbein formalism is superior to the metric formalism for solving Einstein's equations, corresponds to the way actual measurements are made, and is necessary for describing fermions.
  3. Actions: For all the same reasons as for the rest of field theory, gravity should be understood in terms of an action principle, and not just field equations.

Quantum field theory

Some people like to start string theory before finishing a quantum field theory course. That is difficult, and recommended only because of poor course organization: Generally field theory courses end at the end of the second year of graduate study, and research must begin at the beginning of the third year. In any case, all of field theory should be studied, otherwise options in research will be limited. In particular, some modern improvements which are missing in many stale courses are:
  1. Special gauges: The background field gauge, Gervais-Neveu gauge, and (unitary) lightcone gauge introduce useful concepts and calculational advantages.
  2. Two-component spinor indices: "Van der Waerden notation" simplifies calculations not only in supersymmetry but even in QCD, where "spinor helicity" methods (such as the "spacecone" gauge) based on twistors simplify the resulting S-matrix amplitudes.
  3. Color expansion: The "1/N expansion" or "color decomposition" is phenomenological for explaining the relative importance of various hadronic interactions, practical for grouping QCD diagrams in gauge-invariant subsets, and conceptual for describing strings in terms of random lattice models.
  4. Higher spins: Gauge theories of higher spins, including Stückelberg formalisms for massive theories, are useful in understanding string field theory.

Supersymmetry

Supersymmetry is hard to avoid in modern high energy physics. Besides providing solutions to some basic theoretical problems, it is the most common generalization to the Standard Model. However, in spite of its importance, it is often missing from the graduate curriculum.
  1. Superspace: Superspace is the most useful method for treating supersymmetric theories, at both the mechanics and field theory levels, both classical and quantum.
  2. Supergravity: Combining supersymmetry and general relativity leads to supergravity. Again superspace methods are useful, at both the classical and quantum levels.
  3. Higher dimensions: For string theory one needs to understand supersymmetry and supergravity in higher dimensions.

Strings

String theory forms the main portion of research in high energy theory today, but is another topic frequently lacking a course.
  1. First quantization: Quantization of strings in both the lightcone gauge and the conformal gauge (using BRST) is the starting point of string theory.
  2. Conformal field theory: Known string theories are based on systems whose quantum mechanics is mathematically equivalent to that of conformal field theory in two dimensions. In particular, in two dimensions bosonization relates bosonic fields to fermionic ones.
  3. Amplitudes: The study of S-matrix amplitudes reveals the observable distinction between strings and particles.
  4. Superstrings: Combining supersymmetry and strings leads to superstrings. As for particles, supersymmetry improves high-energy behavior and prevents some unphysical features such as tachyons.
  5. Compactification: Although used earlier in gravity and supergravity, the higher dimensionality of known string theories requires compactification or similar methods (such as branes) for elimination of higher dimensions.
  6. Duality: All superstring theories are now thought to be equivalent to each other, and to 11-dimensional supermembrane theory.
  7. Green-Schwarz formulation: The manifestly supersymmetric formulation of superstrings has recently found a new application in the Anti-de-Sitter/conformal-field-theory correspondence.
  8. String field theory: The field theory of strings has also made a comeback, now for the study of vacuua not seen in first-quantized approaches.
  9. Random lattices: The quantization of string theory on random worldsheet lattices in terms of fields carrying a confined symmetry is a promising approach that has not been fully explored.

References

Free, by me

  1. Fields is, of course, my recommendation for field theory. Other textbooks are also useful, but this one includes general relativity, an introduction to supersymmetry, and brief introductions to supergravity and strings, as well as all the "modern" field theory topics listed above.
  2. Superspace, which I co-authored with Jim Gates, Marc Grisaru, and Martin Roček, is the best reference on that topic, including (four-dimensional) supersymmetry and supergravity, and quantization of both.
  3. Introduction to string field theory includes an introductory treatment of strings.

Expensive

  1. Superstring Theory (2 volumes), by Green, Schwarz, and Witten, has a more thorough treatment of string theory, especially compactification.
  2. String Theory (also 2 volumes), by Polchinski, is a more recent thorough string theory book, including branes and duality.