  We consider a gauge symmetry in a quantum Hilbert space. The symmetry leads
to that of the heat-kernel and of the anomaly formulae which were previously
obtained by the authors. This greatly simplifies and clarifies the structure of
the formulae. We explicitly obtain the anomaly formulae in two and four
dimensions, which ``unify'' all kinds of anomaly. The symmetry corresponds to
that of the counterterm formulae in the background field method. As an example,
the non-abelian anomaly is considered.
