  In this paper the Lifshitz formula for the Casimir energy between two
dielectrics in zero temperature is derived using box renormalization. Although
there are several derivations for the force in this case in the literature,
including Lifshitz's own proof, so far there has been no unambiguous and
rigorous derivation for energy that we are aware of. Since the energy becomes
important in some cases, e.g. calculation of entropy or heat capacity, using
the correct and precise definition of the Casimir energy, for the first time,
we remove all of the infinities systematically without any ambiguity. This
proof also shows the strength and accuracy of the box renormalization scheme.
