Using Ashtekar variables, we analyze Lorentzian and Euclidean gravity in vacuum up to a constant conformal transformation. Keeping unaltered the symplectic structure in the full theory of complex gravity, 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. This conclusion also holds in the geometrodynamic formulation. As a consequence, it seems impossible to obtain Lorentzian physical predictions from the quantum theory constructed with the Euclidean reality conditions.
Ashtekar variables in classical general relativity