An essential assumption in the Vafa and Witten's theorem on P and CT realiz
ation in vector-like theories concerns the existence of a free energy densi
ty in Euclidean space in the presence of any external hermitian symmetry br
eaking source. We show how this requires the previous assumption that the s
ymmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is
plausible, actually a theorem is still lacking.