  The generalised Wick transform discovered by Thiemann provides a
well-established relation between the Euclidean and Lorentzian theories of
general relativity. We extend this Thiemann transform to the Ashtekar
formulation for gravity coupled with spin-1/2 fermions, a non-Abelian
Yang-Mills field, and a scalar field. It is proved that, on functions of the
gravitational and matter phase space variables, the Thiemann transform is
equivalent to the composition of an inverse Wick rotation and a constant
complex scale transformation of all fields. This result holds as well for
functions that depend on the shift vector, the lapse function, and the Lagrange
multipliers of the Yang-Mills and gravitational Gauss constraints, provided
that the Wick rotation is implemented by means of an analytic continuation of
the lapse. In this way, the Thiemann transform is furnished with a geometric
interpretation. Finally, we confirm the expectation that the generator of the
Thiemann transform can be determined just from the spin of the fields and give
a simple explanation for this fact.
