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):
- Local symmetries: coordinate transformations, local
Lorentz transformations, local Weyl scale transformations.
- 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.
- 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:
- Special gauges: The background field gauge,
Gervais-Neveu gauge, and (unitary)
lightcone gauge introduce useful concepts and calculational advantages.
- 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.
- 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.
- 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.
- Superspace: Superspace is the most useful method for treating
supersymmetric theories, at both the mechanics and field theory levels, both
classical and quantum.
- Supergravity: Combining supersymmetry and general relativity
leads to supergravity. Again superspace methods are useful, at both the
classical and quantum levels.
- 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.
- First quantization: Quantization of strings in both the lightcone
gauge and the conformal gauge (using BRST) is the starting point of string
theory.
- 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.
- Amplitudes: The study of S-matrix amplitudes reveals the observable
distinction between strings and particles.
- Superstrings: Combining supersymmetry and strings leads to
superstrings. As for particles, supersymmetry improves high-energy behavior
and prevents some unphysical features such as tachyons.
- 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.
- Duality: All superstring theories are now thought to be
equivalent to each other, and to 11-dimensional supermembrane theory.
- Green-Schwarz formulation: The manifestly supersymmetric
formulation of superstrings has recently found a new application in
the Anti-de-Sitter/conformal-field-theory correspondence.
- 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.
- 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
- 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.
- 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.
- Introduction to string field theory
includes an introductory treatment of strings.
Expensive
- Superstring Theory (2 volumes), by Green, Schwarz, and Witten,
has a more thorough treatment of string theory, especially compactification.
- String Theory (also 2 volumes), by Polchinski,
is a more recent thorough string theory book, including branes and duality.