  In various background independent approaches, quantum gravity is defined in
terms of a field propagation kernel: a sum over paths interpreted as a
transition amplitude between 3-geometries, expected to project quantum states
of the geometry on the solutions of the Wheeler-DeWitt equation. We study the
relation between this formalism and conventional quantum field theory methods.
We consider the propagation kernel of 4d Lorentzian general relativity in the
temporal gauge, defined by a conventional formal Feynman path integral, gauge
fixed a\' la Fadeev--Popov. If space is compact, this turns out to depend only
on the initial and final 3--geometries, while in the asymptotically flat case
it depends also on the asymptotic proper time. We compute the explicit form of
this kernel at first order around flat space, and show that it projects on the
solutions of all quantum constraints, including the Wheeler-DeWitt equation,
and yields the correct vacuum and n-graviton states. We also illustrate how the
Newtonian interaction is coded into the propagation kernel, a key open issue in
the spinfoam approach.
