The no boundary proposal and quantum gravity

[Morbert]

TL;DR Summary

This is a continuation of a side discussion from a recently closed thread

This is not what the no boundary proposal says. It says that there is no boundary at the beginning of the universe; that means no "single point". It means the 4-D geometry of the universe at the beginning is smooth and geodesically complete and the curvature is finite everywhere, instead of the 4-D geometry being geodesically incomplete and the curvature increasing without bound as a past boundary is approached.

The paper you cited is not a paper about the no boundary proposal, but about a different proposal that is part of an attempt to develop a theory of quantum gravity.

It is cited here, and here like so:

But then what should the conditions be at the ends of space and time? Here J. Hartle and S. Hawking made a suggestion that is as radical as it is elegant [1, 2]: they proposed that there should be no such ends! In other words, they proposed that space and time should have no boundary to our past

Two leading proposals for special quantum states of the universe are the Hartle-Hawking ‘no-boundary’ proposal [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]

And here

We calculate the probability measure on classical spacetimes predicted by the no-boundary wave function (NBWF) [1]

[edit] - and here

Considering the quantum fluctuations inherent to the universe, it is plausible that the universe originated from nothing devoid of any space-time. This idea is a cornerstone of quantum cosmology, with a long history dating back to Lemaitre [1]. The most robust formulations of this idea, such as the no-boundary proposal [2]

The full quote from the paper I cited:

The Euclidean four-geometries summed over must have a boundary. on which the induced metric is h,j. The remaining specification of the class of geometries which are summed over determines the ground state. Our proposal is that the sum should be over compact geometries. This means that the Universe does not have any boundaries in space or time (at least in the Euclidean regime) (cf. Ref. 3). There is thus no problem of boundary conditions. One can interpret the functional integral over all compact four-geometries bounded by a given three-geometry as giving the amplitude for that three-geometry to arise from a zero three-geometry, i.e., a single point.

I don't see how this paper is not a paper about the no boundary proposal.

 

[PeterDonis]

I don't see how this paper is not a paper about the no boundary proposal.

I can only see the abstract because the paper is paywalled. So I can't see the context of your full quote. I agree that "does not have any boundaries in space or time (at least in the Euclidean regime)" looks more like the no boundary proposal, yes. I'll post a clarification in the other thread.

However, the full quote also says "compact four-geometries bounded by a given three-geometry", which doesn't really make sense; "bounded" seems inconsistent with "does not have any boundaries in space or time". I know that the proposal is using the "Euclidean regime", which brings in a whole other set of issues, so it might be that the full manifold being used has a boundary, but only a portion of it actually describes the universe we observe, and that portion has no boundary (though I'm not sure how that would work either).

In short, you were definitely right to mark this thread as "A" level.

 

[PeterDonis]

Moderator's note: Thread moved to the Beyond the Standard Models forum.

 

[PeterDonis]

It is cited here

Figure 13, right panel, of this paper looks like my understanding of the no boundary proposal. Note that there is, well, no boundary.

The "South Pole" point in the diagram, as I understand it, is not an "initial singularity" or "zero dimensional point that turned into the universe". It is part of the "Euclidean regime" (the whole "cup" that is in the right panel but not the left panel of that figure is that regime), in which, heuristically, everything is spacelike and there is no "time" dimension at all.

 

[mitchell porter]

the full quote also says "compact four-geometries bounded by a given three-geometry", which doesn't really make sense; "bounded" seems inconsistent with "does not have any boundaries in space or time". I know that the proposal is using the "Euclidean regime", which brings in a whole other set of issues

The "given three-geometry", is a state of the universe on some spacelike slice.

Suppose you had a theory of quantum gravity, and you wanted the probability amplitude to go from three-geometry A to three-geometry B. One way you could do it, is as a path integral over 4-geometries which had A as their boundary in the past, and B as their boundary in the future.

In this paper, they consider path integrals over 4-geometries where there's only one 3-geometric boundary. The other side is just compact. It's the 4d analogue of a hemisphere, like half a ping-pong ball. The place where it's sliced open is the boundary 3-geometry, and the curved part is the compact "no-boundary initial condition".

They are suggesting that this is a recipe to obtain apriori probabilities for possible 3-geometries, from a quantum cosmology. You are to think of 3-geometry A as complete nothingness, no space or time... You could calculate the most likely 3-geometries, and compare them with the observed state of the universe.

The problem is, their path integral is a sum over Euclidean geometries, whereas the history of the universe, at least the part we know about, has been Lorentzian in signature. They do some kind of analytic continuation in the paper, but it has never been clear to me, whether the no-boundary condition should itself be considered solely an artefact of the analytic continuation. They talk about it as if it's "real", but then it needs to apply in the Lorentzian picture as well.

 

[Morbert]

The problem is, their path integral is a sum over Euclidean geometries, whereas the history of the universe, at least the part we know about, has been Lorentzian in signature. They do some kind of analytic continuation in the paper, but it has never been clear to me, whether the no-boundary condition should itself be considered solely an artefact of the analytic continuation. They talk about it as if it's "real", but then it needs to apply in the Lorentzian picture as well.

One line of research has been a formulation of the proposal using a Lorentzian path integral rather than a Euclidean one, but it runs into issues (see here and here).

Sberna et al have claimed to resolve unstable perturbation issues by fixing the initial momentum rather than the original size, so that geometries with nonzero initial sizes are included. But this reintroduces singularity issues.

As far as I can tell the proposal is of interest but ultimately underdeveloped.

 

[PeterDonis]

In this paper, they consider path integrals over 4-geometries where there's only one 3-geometric boundary.

That would be basically the 3-geometry of our universe "now"?

 

[mitchell porter]

That would be basically the 3-geometry of our universe "now"?

In theory you could think like that. But in practice I think these quantum cosmological ansatze really only get applied to the very very early universe, in order to get predictions for the state of the universe at the beginning of the cosmologically classical regime we now live in.