Musings

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

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

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

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

is unitary.

But, say Gottesman and Preskill, the Hamiltonian governing the time-evolution inside the horizon is unlikely to be diagonal $H \neq H_M\otimes 1 + 1\otimes H_{\mathrm{in}}$. 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 $T$, the transition amplitude is given by

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

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

Perhaps one can compensate for the effect of interactions by adjusting the choice of final state $\langle F |$, 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.