We derive the fixed-$\Lambda$ and unimodular propagators using the path
integral formalism as applied to the Einstein-Cartan action. The simplicity of
the action (which is linear in the lapse function) allows for an exact
integration starting from the lapse function and the enforcement of the
Hamiltonian constraint, leading to a product of Chern-Simons states if the
connection is fixed at the endpoints. No saddle point approximation is needed.
Should the metric be fixed at the endpoints, then, depending on the contour
chosen for the connection, Hartle-Hawking or Vilenkin propagators are obtained.
Thus, in this approach one trades a choice of contour in the lapse function for
one in the connection, where appropriate. The unimodular propagators are also
trivial to obtain via the path integral, and the previously derived expressions
are recovered.
