  Homogenization theory is used to calculate the macroscopic dielectric
constant from the quantum microscopic dielectric function in a periodic medium.
The method can be used to calculate any macroscopic constitutive relation, but
it is illustrated here for the case of electrodynamics of matter. The so-called
cell problem of homogenization theory is solved and an explicit expression is
given for the macroscopic dielectric constant in a form akin to the
Clausius-Mossotti or Lorentz-Lorenz relation. The validity of this expression
is checked by showing that the macroscopic dielectric constant is causal and
has the expected symmetry properties, and that the average of the microscopic
energy density is the macroscopic one. Finally, the general expression is
applied to Bloch eigenstates. Finally, the corresponding many-body problem is
briefly discussed.
