Thiemann transform for gravity with matter fields

The generalized Wick transform discovered by Thiemann provides a well estab lished relation between the Euclidean and Lorentzian theories of general re lativity. 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 m atter phase space variables, the Thiemann transform is equivalent to the co mposition of an inverse Wick rotation and a constant complex scale transfor mation of all fields. This result also holds for functions that depend on t he shift vector, the lapse function and the Lagrange multipliers of the Yan g-Mills and gravitational Gauss constraints, provided that the Wick rotatio n 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 tran sform can be determined just from the spin of the fields and give a simple explanation for this fact.