  zeta-regularized traces, resp. super-traces, are defined on a classical
pseudo-differential operator A by: tr^Q(A):= f.p.tr(A Q^{-z})_{|_{z=0}}, resp.
str^Q(A):= f.p.str(A Q^{-z})_{|_{z=0}}, where f.p. refers to the finite part
and Q is an (invertible and admissible) elliptic reference operator with
positive order. They are widly used in quantum field theory in spite of the
fact that, unlike ordinary traces on matrices, they are neither cyclic nor do
they commute with exterior differentiation, thus giving rise to tracial
anomalies. The purpose of this article is to show, on two examples, how tracial
anomalies can lead to anomalous phenomena in quantum field theory.
