In ordinary quantum mechanics we work in euclidean spacetime or in a Minkowski space. But in quantum gravity, the physical process may result in a spacetime geometry which changes, and significantly differs, from these familiar geometries. A black hole may form, for instance.
A classical analogue is a rubber membrane. If the membrane is approximately planar, then we can define the interference pattern of waves in the membrane in a reasonable way. Simply sum the waves.
But suppose that the geometry of the membrane can change during a physical process. The plane may become a torus, for example. How can we calculate an interference pattern for fundamentally different geometries? How to sum a wave within a plane with a wave within a torus?
A possible solution
The Afshar experiment shows that different paths in a path integral can interact with each other in the course of the experiment. We must be able to calculate intermediate interference patterns to determine the fate of the experiment.
However, radically different spacetime geometries correspond to macroscopically different distributions of mass-energy in spacetime. The interference between macroscopically differing branches of history is negligible.
A tentative solution to the problem: we can (and should) sum waves for branches of the path integral if the branches only differ microscopically. This is how we do everyday quantum mechanics. Simply assume that the change in the geometry of spacetime is negligible.
On the other hand, if the branches differ macroscopically, then we ignore interference effects between them. This is how we do classical mechanics.
The solution is not beautiful. We establish an ad hoc border between the classical world and the quantum world.
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