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December 1, 2003

Gottesman & Preskill

Dan Gottesman and John Preskill have a nice little paper which seems to be a quite devastating critique of a recent proposal by Horowitz and Maldacena on blackhole information.

The blackhole information paradox, in a nutshell, is the apparent difficulty in finding a unitary evolution from the initial state of the infalling matter, which forms the blackhole, to the final state of the outgoing Hawking radiation. Horowitz and Maldacena’s proposal involves a particularly clever decomposition of the Hilbert space of the stuff “inside” the horizon, and the imposition of a final state condition at the blackhole singularity.

Let M\mathcal{H}_M be the Hilbert space of the infalling matter, in\mathcal{H}_{\mathrm{in}} and out\mathcal{H}_{\mathrm{out}} be the Hilbert spaces of the infalling and outgoing Hawking radiation, respectively. Each of these Hilbert spaces are purported to be of dimension N=e SN=e^S, where SS is the entropy of the blackhole. We can take them to have orthonormal bases |e i| e_i \rangle, |f i| f_i \rangle and |g i| g_i \rangle, respectively. The Unruh state, |𝒰 in out| \mathcal{U} \rangle \in \mathcal{H}_{\mathrm{in}}\otimes \mathcal{H}_{\mathrm{out}} of the Hawking radiation is

(1)|𝒰=1N i=1 N|f i|g i | \mathcal{U} \rangle = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} | f_i \rangle \otimes | g_i \rangle

Horowitz and Maldacena propose that the final state, F| M in\langle F | \in \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}}, at the singularity be given by

(2)F|=1N i=1 Ne i|f i| \langle F | = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} \langle e_i | \otimes \langle f_i |

In that case, the evolution operator, T: M inT: \mathcal{H}_M\to \mathcal{H}_{\mathrm{in}} is determined by

(3)T^=F|𝒰=1N i=1 N|g ie i| \hat{T} = \langle F | \mathcal{U} \rangle = \frac{1}{N} \sum_{i=1}^N | g_i \rangle \langle e_i |

with the important proviso that we should consider only conditional probabilities: given that the final state at the singularity is F| \langle F |, what is the amplitude to go from some initial state in M\mathcal{H}_M to some final state in out\mathcal{H}_{\mathrm{out}}? That has the effect of multiplying T^\hat{T} by NN, with the result that

(4)T= i=1 N|g ie i| T= \sum_{i=1}^N | g_i \rangle \langle e_i |

is unitary.

But, say Gottesman and Preskill, the Hamiltonian governing the time-evolution inside the horizon is unlikely to be diagonal HH M1+1H inH \neq H_M\otimes 1 + 1\otimes H_{\mathrm{in}}. Instead, interactions between the infalling Hawking fluctuations and the infalling matter will produce some more general state of the stuff inside the horizon. Instead of TT, the transition amplitude is given by

(5)T˜=NF|U|𝒰 \tilde{T} = N \langle F | U | \mathcal{U}\rangle

for some unitary operator U: M in M inU: \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}} \to \mathcal{H}_M \otimes \mathcal{H}_{\mathrm{in}}.

Depending on the nature of UU, we could have no information loss (T˜\tilde{T} unitary) or total information loss ( the final state of the Hawking radiation being totally independent of the initial state in M\mathcal{H}_M) or anything in between.

Perhaps one can compensate for the effect of interactions by adjusting the choice of final state F|\langle F |, but it’s not exactly clear how one is supposed to do that.

The one thing that String Theory definitely doesn’t hand us is a description of quantum gravitational processes (like the formation and subsequent evaporation of a blackhole) in terms of Hilbert spaces, much less finite-dimensional ones which can be decomposed in the fashion described above. Maybe that’s a sign we need to get beyond our silly attachment to such ideas.

But I don’t know how to do that either.

Posted by distler at December 1, 2003 1:53 AM

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Re: Gottesman & Preskill

In retrospect, the business of imposing a final state boundary condition (‘postselection’) on an arbitrary collection of collapsing matter seems pretty damn suspicious.

I seem to remember Tipler saying something vaguely similar at the Cambridge M-theory Cosmology conference (for quite different reasons, though). The upshot was that the collapsing matter at the end of the (closed) Universe would have to reach the Omega Point in order to rearrange itself (with infinite intelligence and potency) into the unique state for which the final collapse was allowed. Bunkum, of course.

Although, in the face of the intractable difficulties of black hole singularities and information loss, we should try everything once.

If there is to be a final state of gravitational collapse, surely a physically reasonable one would be a state of maximal entropy. But that’s as far from solving the information problem as it can be, and may conflict with the string-theoretic picture of entropy living ‘around the horizon’.

What’s to stop us bringing a collection of collapsing matter in from asymptotically flat past infinity and watching the radiation come out at asymptotically flat future infinity? Could string theory, in principle, give us this S-matrix? (I use ‘in principle’ in its loosest sense…) Or is there some fundamental obstruction there?

Posted by: Thomas Dent on December 3, 2003 2:38 PM | Permalink | Reply to this

Re: Gottesman & Preskill

What’s to stop us bringing a collection of collapsing matter in from asymptotically flat past infinity and watching the radiation come out at asymptotically flat future infinity? Could string theory, in principle, give us this S-matrix? (I use ‘in principle’ in its loosest sense) Or is there some fundamental obstruction there?

In principle, string theory does give us the S-matrix for this process.

In practice, in asymptotically-flat space, string perturbation theory (still our main tool for doing actual calculations) breaks down when one tries to calculate a process of this sort.

In asymptotically-anti-de Sitter, things are, in principle better, as we have a nonperturbative formulation (in terms of a gauge theory living on the conformal boundary of anti-de Sitter space). There, one still has the challenge of doing explicit calculations in the strongly-coupled gauge theory, but all indications are that anti-de Sitter blackholes behave in a perfectly unitary fashion.

Posted by: Jacques Distler on December 4, 2003 8:24 AM | Permalink | Reply to this

Re: Gottesman & Preskill

I should know where to look up this stuff, but I’m asking anyway - Do the AdS black holes have singularities? And what do the singularities look like in gauge theory?

I’ve heard talks about the bulk black hole corresponding with finite temperature field theory but the singularity didn’t seem to get a mention.

Posted by: Thomas Dent on December 5, 2003 12:46 PM | Permalink | Reply to this

Blackhole Singularity in AdS/CFT

Here’s one attempt to probe the singularity of an AdS blackhole from the gauge theory.

I’d meant to blog about that paper sometime. Maybe this’ll be an opportunity to kick-start such a discussion.

Posted by: Jacques Distler on December 5, 2003 1:55 PM | Permalink | Reply to this

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