Quantum mechanics, as for example in the case of a non-relativistic particle, can be treated in either of two ways. One can work with the differential-equation form of the theory, by studying the Schrödinger equation. Alternatively, one can study the Feynman path integral, which gives the integral form of the Schrödinger differential approach. The Feynman path integral has the advantage of incorporating the boundary conditions on the particle, for example that the particle is at spatial position xa at an initial time ta, and at position xb at final time tb. The path integral leads naturally to a semi-classical expansion of the quantum amplitude, valid asymptotically as the action of the classical solution of the equations of motion becomes large compared to Planck's constant ħ.

One moves from quantum mechanics to quantum gravity by replacing the spatial argument x of the wave function by the three-dimensional spatial geometry hij(x). A typical quantum amplitude is then the amplitude to go from an initial three-geometry hijI to a final geometry hijF, specified (say) on identical three-surfaces ΣI, ΣF. To complete the description in the asymptotically flat case, one needs to specify asymptotic parameters such as the time T between the two surfaces, measured at spatial infinity. To make the classical boundary-value problem elliptic and (one hopes) well-posed, one rotates to imaginary time –iT. The Feynman path integral would again give a semi-classical expansion of the quantum amplitude, were it not for the infinities present in the loop amplitudes.