Thursday, May 10, 2018

Chronological sins in the bulk

A week ago, I discussed an Indian paper criticizing the fuzzballs but I neglected a very interesting yet much shorter, 5-page-long paper
On the interior geometry of a typical black hole microstate
by de Boer, van Breukelen, Lokhande, K. Papadodimas, and Erik Verlinde. I've had some interesting exchanges with Kyriakos.

What they do is to try the counterpart of Gao-Jafferis-Wall – but in the case of one CFT only. Two years ago, Gao-Jafferis-Wall considered an AdS-based BTZ black hole with two sides, two identical CFTs sit on the boundary, and they deform their Hamiltonian by a "coupled" double-trace operator which makes the wormhole traversable.

OK, de Boer et al. do the same thing except that they only have one CFT. Correspondingly, they must replace one of the factors in the "double trace deformation" by new operators constructed from the same single CFT. They use the tilded "mirror operators" \(\tilde{\mathcal O}\) discussed in papers by Papadodimas and Raju – also analyzed in several TRF blog posts.

One problem is that they've never shown that these operators may be determined in any unique way – there can arguably be infinitely many solutions for "what can be called a mirror operator". But I will ignore this problem in the rest of this text – nevertheless, the non-uniqueness of answers is an issue you shouldn't quite overlook.

The insertion of \({\mathcal O}\tilde{\mathcal O}\) operators creates shock waves which allow you to extract the information from the previously doomed black hole interior. And they argue that correlators involving the operators \({\mathcal O},\tilde{\mathcal O}\) in the CFT may be shown to agree with some bulk expectations in the bulk with a Penrose diagram that contains both "two sides" of the black hole as well as the "white hole" in the distant past.

So something seems to be right – there's some evidence that a more complicated bulk geometry is dual to the typical microstate in the presence of the (somewhat contrived and not very explicit) deformation that includes the tilded operators. It looks nice except that I still think that what they offer is just some limited positive evidence in favor of a "big package" of claims that should be discussed separately. I think that the evidence in the paper isn't sufficient to accept every piece of the "package" separately, and those pieces of the package shouldn't be conflated.

First of all, while the solution to Einstein's equations that look like white holes may be written – they're really just a time reversal of black holes – I think it's right to say that all white holes are always unphysical at the end. They are infinitely unlikely to occur in a real situation. Why? Because all processes involving real-world black holes – including formation (both acquisitions and mergers, to borrow some financial jargon LOL) and black hole evaporation – lead to a strictly increasing entropy. So their time reversal is strictly forbidden in the thermodynamic limit because the entropy cannot decrease by macroscopic amounts.

As I discussed in some previous texts, a black hole microstate should be said to be exactly the same thing as a white hole microstate. However, only the black hole interpretation is legitimate in the real world due to the logical (and thermodynamic) arrow of time. All well-posed problems about probabilities in Nature are about conditional probabilities that start with a well-defined initial state – one that simply has to have a lower entropy than the possible states in the future. The white hole should be assigned the same "maximum" entropy as the black hole at the end of the evolution, so all of its processes violate the second law of thermodynamics. I am sure that my interpretation isn't exceptional – it's surely the consensus of pages on the Internet that white holes would violate the second law.

So even when some limits of correlators work just fine in the presence of white holes, I still think that these correlators can't be given the Born's rule interpretation in a well-defined sequence of events that include measurements of the initial and final states. White holes should remain just some formal solutions, not acceptable spacetime backgrounds around which you may do generic enough experiments.

In this AdS/CFT diagram, the white hole region is called IV. If you assume that the precise microstate may only be prepared by a projection operator constructed out of the two CFT degrees of freedom – the only microscopic definition of the AdS/CFT system that we know – then these CFT projection operators may only influence their causal future, by causality. Those include localized bulk operators but not those in the white hole region IV. The white hole region IV is forbidden for a particular purpose – you can't have controllable initial states prepared for that region because the region IV occurs "before" all the events in the CFTs.

This statement is completely analogous to the statement that you can't escape from the black hole interior and therefore the precise microstate detectable through the black hole interior can't be identified by a precise enough measurement. To measure the precise final state, you need to bring the information about the interior measurements to the boundary CFT – where you have the required unlimited precision of the apparatuses. But nothing can get out of the black hole interior.

So when various AdS and CFT correlators are being discussed, there are two classes of questions. One of them is the mathematical equality between various correlators that may be constructed regardless of their experimental interpretation; and the other is the question what the correlators imply about probabilities of experiments etc. – the truly physical interpretation – and whether such a physical interpretation exists at all.

Even if the mathematics works at the level of correlators, I think that it doesn't follow that the corresponding spacetime backgrounds should be considered physical. In some counting, they may be infinitely unlikely – just like a classical particle sitting at the top of the potential (inverted harmonic oscillator or any other unstable point). A particle may be sitting there – a solution exists. But it's infinitely unlikely that such a solution may materialize in the real world. I think that all Penrose diagrams with white hole regions should be considered analogous to the particle sitting at the top of the potential. It's something that is infinitely unlikely or infinitely hard to prepare. When something works about the mathematics, it's an interesting mathematical fact – but one that doesn't have any direct physical implications.

One implication of this reasoning is that the geometry – and causal structure – of the bulk should be constructed from the detailed microscopic degrees of freedom (e.g. in the boundary CFT) in a way that violates the time-reversal symmetry. You simply want to interpret the microstates as black holes, not as white holes. So the quantum gravity and/or holography must give you some freedom how you choose the spacetime geometry (I think that the freedom is nearly maximal, you may choose almost any background, a principle that I call the background indifference) and you must choose the freedom so that you obey some basic conditions, e.g. the second law of thermodynamics in the bulk. Note that even though the paper by de Boer et al. tries to legitimize the white holes, they admit the violation of the time-reversal symmetry when they draw the shock waves in a way that is clearly time-reversal-asymmetric (Figure 1).

Almost equivalently, they discuss lots of out-of-time-order commutators and correlators in their paper. Their danger is pretty much the same. The point is that you may compute any correlator of any operators, like \[

\bra\chi ABCD\dots \ket \psi,

\] regardless of the localization of the operators \(A,B,C,D\) in space and (especially) time. However, whenever you calculate probability amplitudes that the initial state prepared in some way will behave in a certain way in the future, you need to restrict your attention to correlators where the operators are ordered chronologically! So the earlier operators need to be close to the ket vector \(\ket\psi\). They prepare the initial state, and engineer it, and so on, and the operators at the later time have to be closer to the bra vector \(\bra\chi\) because they represent the final questions posed by another measurement.

So mathematics of a quantum mechanical theory allows you to discuss a huge amount of correlators but only some of them are "physically allowed" because the operators that you sandwiched in between the bra-ket states are properly chronologically ordered. In fact, I believe that this constraint is strong and could be understood as an important principle of quantum gravity – and it could lead to nontrivial consequences, especially because the operators should better be time-ordered according to any "U-dual frame".

Take just T-duality. There's a subtlety with causality that has fascinated me for years but I still don't know what it implies – I just feel that something nontrivial may be derived out of it, maybe even about the most fundamental laws of quantum gravity. Take type II string theory on a circle. Well, there's some sense in which causality holds and the signals can't propagate faster than the speed of light along the circular dimension \(X\).

However, you may also T-dualize the situation and a similar statement holds in that new description: signals shouldn't really propagate faster than light in the direction of the T-dual coordinate \(\tilde X\). In the original T-dual frame with \(X\), this statement says something about the propagation of the information encoded in the relative phases of states with different winding numbers.

Now, should you believe that both causality claims hold at the same moment? Do they hold precisely in some string field theory sense? And if they do, can you assume that both of them hold simultaneously, or is there some uncertainty-principle-like or complementary-like principle that tells you that you should only impose one of these causal restrictions? Note that to discuss the propagation of a wave packet clearly, the packet has to be sort of localized in \(X\). Can it be localized both in \(X\) and \(\tilde X\)? Well, it looks like it can because the momentum and winding are two independent quantum numbers. However, the wound strings with \(w\neq 0\) cannot really be associated with any particular location on the \(X\) circle at all which is an argument that the causal constraints in \(X\) and \(\tilde X\) cannot be imposed simultaneously. On the third hand (I need a mutated person with many hands), some string-field-theory-like causality or locality holds even for the wound strings. What's the final coherent answer to such questions?

And if both of them hold, can't you find infinitely many other U-duality frames that restrict what operators you may write down so that their products are chronologically ordered? And if you can write many such restrictions, aren't they strong enough to fully pinpoint the theory – what kind of observables a quantum theory of gravity may have?

There are lots of questions. This paper by de Boer et al. is interesting but I feel that too many assumptions are being sold as parts of the same package – which is only supported as a whole – while all these interesting questions should be discussed separately. Maybe the answers to some qualitative questions should be negated – and there could still exist a calculation that quantitatively seems to support that edited package.

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