The invariance group of the general relativistic Hamilton-Jacobi equations is shown to be isomorphic to the group of proper canonical mappings of the solutions of the Einstein field equations. The classical general relativistic Hamilton-Jacobi theory is then elaborated to the point where, via the correspondence principle, the preferred complementary concepts of the quantum theory of gravitation may be identified. The classical considerations indicate that the appropriate complementary concepts to be subject to quantization are the equivalence classes of intrinsic spatial geometries which may coexist in the same space-time manifold and the inequivalent extrinsic space-time stackings of these spatial geometries.

• Received 4 December 1967

DOI:https://doi.org/10.1103/PhysRev.170.1195