  We derive the charged current absorption rate of electron and anti-electron
neutrinos in dense matter using a fully relativistic approach valid at
arbitrary matter degeneracy. We include mean field energy shifts due to nuclear
interactions and the corrections due to weak magnetism. The rates are derived
both from the familiar Fermi's Golden Rule, and from the techniques of finite
temperature field theory, and shown to be equivalent. In various limits, these
results can also be used to calculate neutral current opacities. We find that
some pieces of the response have been left out in previous derivations and
their contribution at high density can be significant. Useful formulae and
detailed derivations are presented and we provide a new open source
implementation of these opacities for use in radiation hydrodynamic simulations
of core-collapse supernovae and neutron star mergers.
