  Inspired by Gilkey's invariance theory, Getzler's rescaling method and
Scott's approach to the index via Wodzicki residues, we give a localisation
formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a
class of differential operators acting on sections of a spinor bundle over an
even-dimensional manifold. This formula is expressed in terms of another local
density built from the symbol of the logarithm of a limit of rescaled
differential operators acting on differential forms. When applied to complex
powers of the square of a Dirac operator, it amounts to expressing the index of
a Dirac operator in terms of a local density involving the logarithm of the
Getzler rescaled limit of its square.
