Question on the preservation of information via mapping to free field states

Question

In Hawking's paper, "Breakdown of predictability in gravitational collapse", the crux of Hawking's argument is as follows:

...,one can extend the principle to treatments in which the gravitational field is also quantized by means of Feynman sum over histories. In this one performs an integration (with an as yet undetermined measure) over all configurations of both matter and gravitational fields. The classical example of black-hole event horizons shows that in this integral one has to include metrics in which the interaction region (i.e. the region over which the action is evaluated) is bounded, not only by the initial and final surfaces, but by a hidden surface as well. Indeed, in any quantum gravitational situation there is a possibility of "virtual" black holes which arise from vacuum fluctuations and which appear out of nothing and disappear again. One therefore has to include in the sum over histories metrics containing transient holes, leading either to singularities or to other space-time regions about which one has no knowledge. One therefore has to introduce a hidden surface around each of these holes and apply the principle of ignorance to say that all field configurations on these hidden surfaces are equally probable provided they are compatible with conservation of mass, angular momentum, etc...

The setup for the argument he provides is to define three Hilbert spaces, $H_1$, $H_2$, $H_3$ for the initial, "hidden", and final space which contains all the data for each respective surface. He the defines a tensor $S_{ABC}$ with indices that refer to each space, with states in the each Hilbert space defined as:

$$\xi_C \in H_1$$ $$\zeta_B \in H_2 $$ $$\chi_A \in H_3$$

such that:

$$\sum \sum \sum S_{ABC} \chi_A \zeta_B \xi_C$$

defines the amplitude to have the initial state, final state and the hidden state on the hidden surface. The arguments is given the initial state, one is unable to determine the final state but,

only the element $\sum S_{ABC}\xi_C$ of the tensor product $H_1 \otimes H_2$.

This is the heart of the information loss problem discussed recently by Polchinksi.

Here is where I have an issue. I see the vacuum as a type of black hole. So if one adds another hidden surface into the mix, it is essentially the same as mapping back to some vacuum state. The classical black hole might be distorting the path of particles from initial to final surface, but if I think of the initial and final surfaces as black hole surfaces as well, then it doesn't seem to be too much of a problem.

In the most radical sense, the particles become free particles as far as the rest of the universe is concerned, so there simply isn't any way for them to interact inside the hole and decohere, so whatever information the particle is carrying is perfectly preserved to be spit out again when infalling Hawking radiation "liberates" the information during the evaporation process (e.g. perfectly preserved information of the past can now be passed back to the universe as it is once again allowed to interact with infalling particles, quantum mechanics ensures that state of the outgoing Hawking radiation will be completely consistent). Essentially, I think Hawking's insertion of a interim surface appears to be capricious fiction.

I am curious as to what is wrong with this argument.

UPDATE: I thought a link to Hawking's new paper is a worthwhile addendum.

Answer 1

What's wrong with it is that he is assuming there is a sum over local metrics. The quantum description of gravity is not a path integral over different metrics in string theory, and shouldn't be, because of the paradoxical behavior Hawking is showing.

Unfortunately, in this case, the problem was identified clearly after the fix was already around. Similar considerations led Heisenberg to suggest formulating theories in terms of S-matrix data instead of local field data, and to dispense with local fields. For strong interactions, this wasn't necessary, but Heisenberg's motivations were somewhat gravitational (he was considering what to do if space and time notions break down) and they correctly resolve these inconsistencies.

The formation and evaporation of a virtual black hole is not described by a path integral where you can draw a sphere surrounding the black hole and do the integral external to the sphere, and glue it to the integral inside the sphere. The quantum gravity path integral is over the boundary state (suppose it's an AdS space), and then you don't have a local path integral in the bulk, only on the boundary, and a nonlocal map to reconstruct the bulk spacetime.

In such holographic scenarios, Hawking's arguments can't even be formulated. This is why the path-integral approach to quantum gravity, which is beset by problems anyway, is dead and string theory is alive.

Although it seemed for a long time that loops might work to make sense of the quantum gravity path integral, the type of loop quantization people have fixated on today is equivalent to discrete Regge calculus, and this can't possibly be right.