GENERALIZED SUMS OVER HISTORIES FOR QUANTUM-GRAVITY .1. SMOOTH CONIFOLDS

Quantum amplitudes for euclidean gravity constructed by sums over comp act manifold histories are a natural arena for the study of topologica l effects. Such euclidean functional integrals in four dimensions incl ude histories for all boundary topologies. However, a semiclassical ev aluation of the integral will yield a semiclassical amplitude for only a small set of these boundaries. Moreover, there are sequences of man ifold histories in the space of histories that approach a stationary p oint of the Einstein action but do not yield a semiclassical amplitude ; this occurs because the stationary point is not a compact Einstein m anifold. Thus the restriction to manifold histories in the euclidean f unctional integral eliminates semiclassical amplitudes for certain bou ndaries even though there is a stationary point for that boundary. In order to incorporate the contributions from such semiclassical histori es, this paper proposes to generalize the histories included in euclid ean functional integrals a more general set of compact topological spa ces. This new set of spaces, called conifolds, includes the nonmanifol d stationary points; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einste in conifolds both converge to Einstein conifolds. Consequently, genera lized euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and con ifold histories that approach a stationary point of the Einstein actio n. Therefore sums over conifold histories provide a useful and self-co nsistent starting point for further study of topological effects in qu antum gravity.