The String Coffee Table

Hi Haelfix -

you wrote:

Why is the Euclidean path integral being used in a semiclassical regime?

I wouldn’t put it this way. The Euclidean path integral is supposed by some (few) people to be the correct definition of full quantum gravity, not just the semiclassical limit.

The resoning is that the Euclidean path integral is better defined than the Lorentzian one, and that one hopes that the transition amplitudes it yields are those of Minkowski-signature quantum gravity when analytically continued back to ‘real time’. The problems are

- that not every Riemannian metric has purely-Riemannian sections after complexified to a pseudo-Riemannian metric, so that there is no direct map between the ‘states’ that the Euclidean path integral sums over and the states that we are really interested in.

- that it is not at all clear what Wick rotation is supposed to mean when there is no notion of time. Just consider the case where spacetime topology is nontrivial, which is quite essential for Hawking’s considerations. Then what does it mean to do a Wick-rotation?

Usually, its the late orders of perturbation series that reveal the semiclassical limit

Right now I don’t know what you have in mind here.

You probably know that, but just to make myself clear let me say that the semicalssical limit is usullay defined as the limite where $h\to 0$ asymptotically. This means that in the path integral

contributions from the action $S[\varphi ]$ which vary rapidly with the field configuration $[\varphi ]$ on scales of $\hslash$ tend to cancel each other, due to their quasi-random phases. Hence when expanding $S[\varphi ]$ about a classical solution ${\varphi }_0$ one can write

i.e. expand the action in powers of $h$ and only keep some low orders.

I don’t follow the argument that the nontrivial topologies factor to a constant, or alternately that the nonunitarity goes away asymptotically. Why exactly?

I think that’s the part that would definitely need a real proof. What Hawking does is mentioning a plausibility argument, which uses the fact that, he claims, whenever the topology is nontrivial ‘the fields’ will decay exponentially in the asymptotic future.

I don’t know the details of that, but I got the impression that for the case of ${\mathrm{AdS}}_3$ this is a result by Maldacena, which he briefly mentions somewhere in the second half of the talk.

I don’t get the relevance of ADS/CFT either in this situation.

To my mind AdS/CFT might be the only relevant thing here, to put it bluntly!

If AdS/CFT is right (so far it has been checked only to second order (first stringy contributions) plus maybe a little bit more) then it follows that the stringy UV-completion of supergravity for asymptotically AdS configurations can be mapped in a weird but 1-1 way to a non-gravitational quantum field theory on the boundary of AdS. The important point is that this quantum field theory can be much better understood than the theory in the bulk. In particular we know that the field theory is nicely behaved in that it is unitary. This implies (at least naively) that also the gravitational theory in the bulk is. And this again is precisely the result that Hawking would like to obtain.

Hi Urs, theres one last thing i’ve been puzzling about.

Wick rotation in general is hard to do outside of flat space, as there is no easy notion of time.

I read a note about this on sci.physics.research, where Serenus argues that for consistency in ADS/CFT it requires that such an act be admissible.

Well, one this is not quite satisfying mathematically, and two its not clear how this generalizes at all.

Now, i’ve actually seen some attempts at solving this before. One proposal is to wick rotate on the space of guage fixed metrics (using the proper time guage), as this will not involve a complexification procedure. The drawback is the topology must be fixed, and we are not looking at saddle points of the path integral like Hawkings claims.

But still, its not clear to me how applicable this guage is, as its not covariant. Nor if its relevant to the talk.

In any case, I’d still like to see a sensible argument given to how we can use wick rotation without full time like killing vectors.

Hi Haelfix -

you wrote:

I just don’t see why that should necessarily encompass the bare gravitational theory without anything else, or has that already been shown in the past?

Oh, yes, right. This is precisely a point which I tried to make before. I was surprised that Hawking adopted AdS/CFT reasoning, because one cannot do that without accepting stringy degrees of freedom and all that. But if we do that - what’s the point of arguing non-perturbatively with the Euclidean path integral of pure gravity?! That’s just self-inconsistent reasoning to me…

I agree. Knowing that the gravitational evolution is unitary alone does not tell us how the information is preserved in detail.

Apart from issues with the measurement problem I think one also has to take into account that AdS/CFT predicts stringy degrees of freedom, which might play a crucial role in how information evolves through a black hole.

This is for instance the point of view proposed by Mathur, e.g.:

Samir Mathur: Where are the states of a black hole? (2004)

Abstract:

We argue that bound states of branes have a size that is of the same order as the horizon radius of the corresponding black hole. Thus the interior of a black hole is not `empty space with a central singularity’, and Hawking radiation can pick up information from the degrees of freedom of the hole.