  The Hartle-Hawking state is a proposal for a preferred initial state for
quantum gravity, based on a path integral over all compact Euclidean
four-geometries which have a given three-geometry as a boundary. The wave
function constructed this way satisfies the (Lorentzian) Hamiltonian constraint
of general relativity in ADM variables in a formal sense. In this article we
mimic this procedure of constructing an initial state in terms of
Ashtekar-Barbero variables, and observe that the wave function thus constructed
does not satisfy the Lorentzian Hamiltonian constraint even in a formal sense.
We also investigate this issue for the relativistic particle. We finally
suggest a modification of the proposal that does satisfy the constraint at
least in a formal sense and start to consider its implications in quantum
cosmology. We find that for certain variables, and in the saddle point
approximation, the state is very similar to the Ashtekar-Lewandowski state of
loop quantum gravity.
