Classical Casimir effect: The interaction of ideal parallel walls at a finite temperature

We consider two large ideal parallel plates, each l1 by l2, at a temperature T. The plates are a distance l3 apart, where l3<>(ℎ/2π)c/l3, AT/l1l2 becomes (ℎ/2π) independent, that is, classical. The classical result is given by AClT/l1l2=AT∼∞/l1l2=-ζ(3)kBT/8πl32, where ζ(3)=1.20... . We show that the classical limit can be obtained within the framework of renormalized classical theory, the ultraviolet catastrophe having been eliminated on taking the difference between the electromagnetic fields in the presence and absence of the walls; a Casimir effect could have been obtained before the advent of quantum field theory, or even of quantum mechanics. The domain of validity of the result could not have been obtained, nor would the experimental verification of the theory have been remotely possible