Synopsis for Astronomy 734 "Field Theory for Astronomers"
In this 700 level course I will give an elementary introduction to field
theory for astronomers. Spontaneous symmetry breaking with a massive
scalar field (the Higgs field) provides the foundation for the modern
theories of the electromagnetic and weak interactions. Once released
from the bottle, the genie of the scalar field has been widely embraced
by cosmologists; the `inflaton' is invoked to drive inflation, candidates
for the dark matter include the `axion' and related particles, while the
existence for the unexpected acceleration of the Universe at late times
has led theorists to proposed yet more fields with fanciful names like
`quintessence'. In this course, I will focus mainly on bosonic fields.
These include all of the cosmologically relevant fields, and also the
electromagnetic field. The approach I will take will be to analyze in
some detail a very simple mechanical model composed of beads, rods and
springs (the BRS model) that turns out to be mathematically equivalent
to the fully relativistic scalar field. The advantage of this approach
is that the model can be very simply visualized, and understood (the
course will make use of animations to illustrate the physics) and yet
all of its properties carry over to the physically more abstract fields
invoked my cosmologists.
The outline for the course is as follows:
1) Quick review of the most elementary concepts of Lagrangian dynamics.
The action and the Lagrangian --- the principle of least action ---
energy and momentum conservation. (Ch 2)
1.1) Review of properties of dispersive waves (appendix D)
2) The BRS model --- discrete model --- continuum limit --- covariance
of the BRS model. (Ch 15-16)
3) Conservation laws. Conservation of wave-momentum --- energy and
momentum in the BRS model --- wave-momentum paradoxes --- conservation of
`charge' --- conservation of particle number. (Ch 16)
4) Interacting fields. Simple models for interactions between fields ---
resonance conditions. (Ch 16)
5) The ideal fluid limit of classical wave-mechanics --- the stress
energy tensor and its evolution --- the limits of classical field theory. (Ch 16)
6) Quantum mechanics of fields. The simple harmonic oscillator --- the
Heisenberg, Schroedinger and `interaction' pictures --- the S-matrix ---
free fields --- interactions --- scattering --- self interactions ---
Feynman rules --- kinematic constraints. (Ch 17)
7) Relativistic field theory. The Klein-Gordon field --- quantum
electrodynamics --- connection to kinetic theory and nucleosynthesis. (Ch 18)
8) Scalar fields in cosmology. The scalar field in an expanding universe
--- non-relativistic scalar fields --- chaotic inflation --- fluctuations
from inflation --- self-ordering fields (monopoles, domain walls, cosmic
string, texture). (Ch 18, 29, 32)
After taking this course, the student will be able to read intelligently
any of the many hundreds of cosmology papers each year that invoke
hypothetical fields (the generally accepted rule is that you are only
allowed to invoke the `tooth fairy' once --- per paper!) and will be able to
hold forth with confidence on cosmology and field theory at cocktail parties ;-)
More seriously, the course is also intended to give the student a very basic
grounding in the current understanding of `the way the world works', including
what aspects of the world are fundamentally classical and which require
quantum mechanics? what does momentum conservation mean when applied to
waves? and last, but not least, what is a photon anyway?
Much of the material for the course can be found in chapters 2, 15, 16, 17, 18
29 and 32 of my ``Elements of Astrophysics''. This is available at
http://www.ifa.hawaii.edu/~kaiser/lectures/elements.pdf .