  In this course we give a selfcontained introduction to the quantum field
theory for trapped atomic gases, using functional methods throughout. We
consider both equilibrium and nonequilibrium phenomena. In the equilibrium
case, we first derive the appropriate Hartree-Fock theory for the properties of
the gas in the normal phase. We then turn our attention to the properties of
the gas in the superfluid phase, and present a microscopic derivation of the
Bogoliubov and Popov theories of Bose-Einstein condensation and the
Bardeen-Cooper-Schrieffer theory of superconductivity. The former are
applicable to trapped bosonic gases such as rubidium, lithium, sodium and
hydrogen, and the latter in particular to the fermionic isotope of atomic
lithium. In the nonequilibrium case, we discuss various topics for which a
field-theoretical approach is especially suited, because they involve physics
that is not contained in the Gross-Pitaevskii equation. Examples are quantum
kinetic theory, the growth and collapse of a Bose condensate, the phase
dynamics of bosonic and fermionic superfluids, and the collisionless collective
modes of a Bose gas below the critical temperature.
