  Using Ashtekar variables, we analyze Lorentzian and Euclidean gravity in
vacuum up to a constant conformal transformation. We prove that the reality
conditions are invariant under a Wick rotation of the time, and show that the
compatibility of the algebra of commutators and constraints with the involution
defined by the reality conditions restricts the possible values of the
conformal factor to be either real or purely imaginary. In the first case, one
recovers real Lorentzian general relativity. For purely imaginary conformal
factors, the classical theory can be interpreted as real Euclidean gravity. The
reality conditions associated with this Euclidean theory demand the hermiticity
of the Ashtekar connection, but the densitized triad is represented by an
anti-Hermitian operator. We also demonstrate that the Euclidean and Lorentzian
sets of reality conditions lead to inequivalent quantizations of full general
relativity. As a consequence, it seems impossible to obtain Lorentzian physical
predictions from the quantum theory constructed with the Euclidean reality
conditions.
