  We construct a positive complexifier, differentiable almost everywhere on the
classical phase space of real triads and $SU(2)$ connections, which generates a
Wick Transform from Euclidean to Lorentzian gravity everywhere except on a
phase space set of measure zero. This Wick transform assigns an equal role to
the self dual and anti-self dual Ashtekar variables in quantum theory. We argue
that the appropriate quantum arena for an analysis of the properties of the
Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum
Gravity (LQG) rather than its kinematic Hilbert space. We examine issues
related to the construction, in quantum theory, of the positive complexifier as
a positive operator on this diffeomorphism invariant Hilbert space. Assuming
the existence of such an operator, we explore the possibility of identifying
physical states in Lorentzian LQG as Wick rotated images of physical states in
the Euclidean theory. Our considerations derive from Thiemann's remarkable
proposal to define Lorentzian LQG from Euclidean LQG via the implementation in
quantum theory of a phase space `Wick rotation' which maps real
Ashtekar-Barbero variables to Ashtekar's complex, self dual variables.
