  For various theories, in particular gauge field theories, the algebraic form
of the Hamiltonian simplifies considerably if one writes it in terms of certain
complex variables. Also general relativity when written in the new canonical
variables introduced by Ashtekar belongs to that category, the Hamiltonian
being replaced by the so-called scalar (or Wheeler-DeWitt) constraint. In order
to ensure that one is dealing with the correct physical theory one has to
impose certain reality conditions on the classical phase space which generally
are algebraically quite complicated and render the task of finding an
appropriate inner product into a difficult one. This article shows, for a
general theory, that if we prescribe first a {\em canonical} complexification
and second a $^*$ representation of the canonical commutation relations in
which the real connection is diagonal, then there is only one choice of a
holomorphic representation which incorporates the correct reality conditions
{\em and} keeps the Hamiltonian (constraint) algebraically simple ! We derive a
canonical algorithm to obtain this holomorphic representation and in particular
explicitly compute it for quantum gravity in terms of a {\em Wick rotation
transform}.
