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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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Remember that the sum over histories in the NBWF, in a minimal interpretation, is just a way of constructing solutions to the Wheeler-deWitt equation. Like any differential equation, also the WDW equation needs to be supplemented with boundary conditions to yield unique solutions, but it is not very clear what these should be on abstract superspace (the space of all geometries minus diffeomorphisms). The advantages of a formulation in terms of a path integral are twofold at least; it allows for an attractive interpretation in terms of the sum over histories of ordinary QM, but more importantly it allows one to impose boundary conditions in a much simpler way. There are many boundary conditions that can be chosen still, but Hartle and Hawking's chose that 'There is no boundary'. That is, the only boundary of the 4-manifold is the final surface. This means that the geometry has to be smoothly capped off, and especially at t=0 (time is of course not uniquely defined but you know what I mean). If you want to keep the geometry of the saddle points real, you in general have to go over to Euclidean time to smooth of the singularity. One has to be careful though, these saddle point geometries are not the same as the Lorentzian histories associated to them. the NBWF gives you the Cauchy data on your final surface and then you evolve back in time using the Einstein equations. Many of these histories might correspond to bouncing universes.

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  • $\begingroup$ Ah, OK. So the Euclidean path integral is just a way to get amplitudes for different states. But shouldn't be interpreted as histories? This kind of seems at odds with quantum field theory in which sum over histories is what makes it all consistent. i.e. history from A to B is sum over all intermediate histories passing through C. etc. $\endgroup$ – zooby Mar 12 at 19:16
  • $\begingroup$ @zooby Exactly! Well, I’m also not happy with this from a philosophical point of view but it is not different from the situation in quantum mechanics. In the orthodox interpretation of quantum mechanics the ontology is only the wave function. Particles do not have definite trajectories. Only when measured they have a definite position. So yes, this sum over histories seems to say it takes all possible paths but this is just the same old quantum mechanics interpretation problem. Quantum cosmology is by no ways different in this regard. So if you want to know a real path, you use EFE’s. $\endgroup$ – Michael Angelo Mar 12 at 19:50

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