Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

The algebraic form of the Hamiltonian or Hamiltonian constraint of various (field) theories simplifies considerably if one uses certain complex-valued phase space variables. We show, for a general theory, that if we prescribe first a canonical complexification and second 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 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 Wick rotation transform.