  A simple pedagogical introduction to the effective action method of quantum
field theory is given at a level suitable for beginning postgraduate students.
It is shown how to obtain the effective potential at zero temperature from a
regularized zero-point energy. The results are applicable to curved as well as
to flat space. The generalization to finite temperatures is also given. It is
shown how to obtain high temperature expansions of the thermodynamic potential
for the neutral free Bose gas and the charged Bose gas in both the relativistic
and nonrelativistic limits. The results are obtained for an arbitrary spatial
dimension and in curved space. Results are also obtained for the
self-interacting relativistic gas in three spatial dimensions. A detailed
discussion of how the formalism may be applied to study Bose-Einstein
condensation is given. The interpretation of Bose-Einstein condensation as
symmetry breaking is discussed. Application is given to the study of charged
bosons, both relativistic and nonrelativistic, in a constant magnetic field.
The Meissner effect is obtained for the nonrelativistic gas in three spatial
dimensions. The final application is to the study of nonrelativistic bosons in
a harmonic oscillator confing potential trap. A number of analytical approaches
to this are discussed.
