From tracial anomalies to anomalies in Quantum Field Theory

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.