In Hawking's no boundary proposal he assumed that the the amplitude for a state A (such as a given 3-manifold) is the sum of all amplitudes for histories starting at a special big-bang-state (like the south pole of a sphere) to the state A.

What if we instead we said the amplitude for a state A is the sum of all amplitudes starting at any-possible state B and ending at state A?

Would this make sense?

I think it would look like our Universe if the amplitudes for states starting at a small hot Universe like the Big Bang, outweighed most of the other amplitudes. (Such as starting almost identical to A). But none of these histories would ever have to include a state of zero-size. And you would not need a Euclidean section or imaginary time.

In Hawking's case we would be summing over closed 4-spheres. (Or the square of two hemi-4-spheres). Whereas in the second case we would be summing over $S_1\times S_3$ manifolds. (Or the square of cylinders).

We would have to assume that the amplitude for a state starting and ending at the same state is not simply 1.


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