Nernst's heat theorem for Casimir-Lifshitz free energy

Consideration of the Lifshitz expression for the Casimir free energy on the real frequency axis rather than the imaginary Matsubara frequencies as is customary sheds light on the ongoing debate regarding the thermodynamical consistency of this theory in combination with common permittivity models. It is argued that when permittivity is temperature independent over a temperature interval including zero temperature, a cavity made of causal material with continuous dispersion properties separated by vacuum cannot violate Nernst's theorem (the third law of thermodynamics). The purported violation of this theorem pertains to divergencies in the double limit in which frequency and temperature vanish simultaneously. While any model should abide by the laws of thermodynamics within its range of applicability, we emphasize that the Nernst heat theorem is a relevant criterion for choosing among candidate theories only when these theories are fully applicable at zero temperature and frequency.