  I derive a general effective theory for hot and/or dense quark matter. After
introducing general projection operators for hard and soft quark and gluon
degrees of freedom, I explicitly compute the functional integral for the hard
quark and gluon modes in the QCD partition function. Upon appropriate choices
for the projection operators one recovers various well-known effective theories
such as the Hard Thermal Loop/ Hard Dense Loop Effective Theories as well as
the High Density Effective Theory by Hong and Schaefer. I then apply the
effective theory to cold and dense quark matter and show how it can be utilized
to simplify the weak-coupling solution of the color-superconducting gap
equation. In general, one considers as relevant quark degrees of freedom those
within a thin layer of width 2 Lambda_q around the Fermi surface and as
relevant gluon degrees of freedom those with 3-momenta less than Lambda_gl. It
turns out that it is necessary to choose Lambda_q << Lambda_gl, i.e.,
scattering of quarks along the Fermi surface is the dominant process. Moreover,
this special choice of the two cutoff parameters Lambda_q and Lambda_gl
facilitates the power-counting of the numerous contributions in the
gap-equation. In addition, it is demonstrated that both the energy and the
momentum dependence of the gap function has to be treated self-consistently in
order to determine the imaginary part of the gap function. For quarks close to
the Fermi surface the imaginary part is calculated explicitly and shown to be
of sub-subleading order in the gap equation.
