Clayton informed me that Raphael Bousso who has written the most meaningful reply to the AMPS black hole firewall paradox has completely reverted his position.
I have previously discussed the black hole firewall argument and Raphael's first reply to it.
Now we see two versions of his preprint,
v1: Observer Complementarity Upholds the Equivalence Principle (firewalls are wrong)Can you spot the difference? ;-) Well, I remain an infidel but I think that the second version of Raphael's paper is still pretty clearly written.
v2: Complementarity Is Not Enough (firewalls are true and deep)
At the beginning of v2, Bousso reviews the AMPS paradox. I think that there are many holes in it. But let's start. Most generally, the AMPS argument leads to this paradoxical composition of facts:\[
A\heartsuit B,\\
R_B\heartsuit B.
\] The symbol \(\heartsuit\) means "[near] maximum entanglement" (a form of love) between the two subsystems. It's being said that a system, \(B\), cannot be [near] maximally entangled with two other systems, in this case \(R_B\) and \(A\), so the relationships above can't simultaneously hold and some of the assumptions have to be invalid.
Maximum entanglement means that the entanglement entropy of a subsystem is maximized (equal to \(\ln(N)\) where \(N\) is the dimension of the system's Hilbert space). The entanglement entropy of a subsystem is the von Neumann entropy of its density matrix \[
S = -{\rm Tr}\,\rho_{\rm subsystem} \ln \rho_{\rm subsystem}
\] obtained by tracing over the other subsystem.\[
\rho_{\rm subsystem} = {\rm Tr}_{\rm rest}(\rho_{{\rm subsystem}\otimes {\rm rest}})
\] Clarifying the symbols for the degrees of freedom
There are lots of Hilbert spaces that describe various regions of the spacetime, various subsets of the information available to different observers. Let's look at them.
A long time ago, there was a star in a pure state – by assumption. All observers, whether or not they will later land inside the black hole, should agree about the purity of the state. I may challenge this point later but let's continue with the presentation by Bousso.
So this star in the pure state evolves, collapses in a large black hole, eats the stuff around it, and finally completely evaporates into the Hawking radiation. It takes a lot of time for it to radiate. There is some point in the middle at which the black hole is already shrunk so that its entropy is one-half of the original full value.
At this point, the black hole becomes "old". The spacetime around it resembles the Schwarzschild solution and we will choose \(t\), the time coordinate in the Schwarschild's original solution (or another similar solution), to be \(t=0\) at the moment that divides the "young black hole era" from the "old black hole era". It's some spatial slice. The sketchiness of the explanation what \(t\) exactly is – how the slice is obtained – suggests that it shouldn't matter but one should recheck all these things, too.
At any rate, at \(t=0\), there are two observers, Alice and Bob. Alice will jump into the black hole, Bob will stay out. Bob will be able to witness the long life of the Universe and see that the original pure state evolves into a pure state of the Hawking radiation, \(R R'\). The black hole may have been present for a long time but it was just an episode in a unitary scattering process. The Hilbert space for the full Hawking radiation \(RR'\) is composed of the "early radiation" \(R\) that already exists somewhere in the space at \(t=0\) and the "late radiation" \(R'\) that will be created for \(t\gt 0\) and is inaccessible to Alice.
At \(t=0\), the information that will be reprinted in the late radiation \(R'\) is stored in the combined system \(BH\) which sort of sounds like the acronym for the black hole, and it is the black hole from Bob's viewpoint but it is not an acronym. Here \(B\) are some "near horizon modes" at distances comparable to the black hole radius from the horizon; \(H\) is the stretched horizon, a very thin layer above the actual event horizon.
For Alice, the vicinity of the event horizon looks like a regular place of the spacetime. The event horizon itself is approximately a Rindler horizon. Her spacetime is divided into left and right (external and internal) wedges. The space over there, in this old black hole, is almost empty, and this near Minkowski space is a maximally entangled state mixing states in the left wedge and the right wedge.
If \(A\) denotes the "inner side" mirror of the region \(B\) described above, the infalling babe Alice sees that \(A\heartsuit B\). On the other hand, as the external guy Bob may conclude, we're told, that \(R_B\heartsuit B\) where \(R_B\) is a part of the early radiation \(R_B\subset R_{BH} \subset R\) that purifies \(B\subset BH\) much like \(R_{BH}\subset R\) purifies \(R'=BH\).
(The proposition "Bush purifies Reagan" means "while Reagan may be in a mixed state himself, when you consider Reagan+Bush, their total state is already pure". Purification is the constructive proof of the point that every density matrix may be obtained by tracing over some extra degrees of freedom.)
In other words, it's been previously argued that the early radiation is maximally entangled with the late radiation, \(R\heartsuit R'\) – this claim should be rechecked again, of course – and because the (macroscopic) near-horizon region \(B\) is effectively embedded in the late Hawking radiation \(R'\), by the evolution, \(B\) must be maximally entangled with some tensor factor of the early radiation, too.
There are lots of points to be checked, points that are suspicious, and so on. For example, the "approximate maximum entanglements" may need some very careful quantification because we're dealing with entropies of vastly different magnitude. In particular, when the low-energy infalling observer Alice says that \(A\heartsuit B\), she only talks about some low-energy degrees of freedom in \(A\) and \(B\) – modes with low enough energy so that they can't disturb the predetermined geometry near the Rindler wedge – and these modes carry entropy that is vastly smaller than the black hole entropy, and its counterparts in \(R'\) may have been neglected by the approximations.
Nevertheless, Raphael's main conclusion is still the same AMPS message: "The field theory degrees of freedom just inside and outside the black hole cannot be mutually entangled or even correlated."
Why Bousso got converted?
AMPS (especially Don Marolf) have previously convinced Daniel Harlow by an argument that is sketched in Bousso's v2 paper, too.
Newborn firewall believer Bousso describes his infidelity in his previous life – the "things are OK due to observer complementarity" – as follows:
Alice, on the other hand, jumps into the black hole and thus cannot measure theNow, the believer Raphael's reply to this previous sin looks like this:
late Hawking radiation, \(R'\). Therefore, she cannot verify Eq. (1.1), \(R\heartsuit R'\), directly, and thus establish a conflict between it and Eq. (1.3), \(A\heartsuit B\). She can experience the vacuum at the horizon, but by then it is too late to tell Bob, or (equivalently) to fire her rockets and become like Bob, a distant observer at late times.
In order for this resolution to be valid, it must pass a consistency check articulated by Harlow [20]: it must be impossible for Alice to measure \(B\) before she reaches the stretched horizon. Otherwise, Alice could measure the relevant subset of the late Hawking radiation on her way to the horizon, in its incarnation as the near-horizon modes \(B\subset BH = R'\). At this point she could still decide to fire her rockets and stay outside, so her theory must agree with Bob's theory. It must predict that \(B\) is maximally entangled with the early radiation, Eq. (1.4). But then her theory cannot also predict that \(B\) is maximally entangled with the modes inside the horizon, Eq. (1.3). If Alice can measure \(B\) before crossing the horizon, then complementarity is not enough to evade the firewall argument.Bousso claims that for large enough black holes, a sufficient number of qubits may be extracted to make the contradiction real even in the absence of perfect measurements and he continues by a discussion of other "radical solutions" to the AMPS paradox.
A widespread misunderstanding of quantum mechanics
However, the long paragraph I just quoted – the confession of Raphael's sins – contains what I consider the most obvious bug in Raphael's v2 paper which is probably shared by AMPS, Harlow, and everyone else who has agreed that there exists a firewall paradox.
We're told that shortly before \(t=0\), Alice could still agree to marry Bob, join his spaceship, and escape the black hole, so she must have the "same theory" for the late Hawking radiation. First, it's very strange that Bousso is using the word "theory". He doesn't really mean a theory (like GR or string theory). He means a description of the physical state describing the Universe.
But what's the problem here? It's just not true that Alice must have a "theory" for the late Hawking radiation. She will never measure it – by the definition of Alice – so what should be properly called the theory – quantum mechanics with a particular Hamiltonian or whatever defines the evolution – doesn't have to answer and actually doesn't answer what these non-existing measurements will do.
Quantum mechanics only predicts probabilities of experiments that actually may be done – or "will be done", if I make the point even sharper. Alice can't simultaneously measure properties of the late Hawking radiation \(R'\) as well as some internal modes slightly inside the black hole, \(A\), so quantum mechanics doesn't have to make predictions for these mutually excluding histories at the same moment!
In particular, there's also absolutely nothing that invalidates the "old" interpretation of the black hole complementarity which is that the degrees of freedom in \(A\) are equivalent to a subset of the degrees of freedom in \(R'\). So \(B\) can be maximally entangled with both of them – because they're the same degrees of freedom. Depending on whether or not Alice marries Bob, she will use two different states to describe and predict her future measurements. She will either use Bob's description in which \(A\) disappears and only the "external parts of the black hole", \(B\) and \(H\), produce degrees of freedom that are appearing in \(R'\); or she will jump inside the black hole so that \(A\) become some of her degrees of freedom but \(R'\) will be unphysical for her. It's the point of complementarity – even in Bohr's sense – that no one can talk about \(A\) and \(R'\) at the same moment.
The fact that Alice faced a choice before \(t=0\) doesn't mean that she must have "theories" for both possible continuations of her life. Her brain at that moment was in a Schrödinger cat-like superposition (or density matrix) that had both possibilities: she would either say Yes or No to Bob's proposal to marry.\[
\ket{\text{Alice's brain}} = a_{\rm Yes}\ket{ {\rm Yes}}+a_{\rm No}\ket {{\rm No}}.
\] The probabilities are given by \(|a_i|^2\). But the two pieces of this wave function mean "EITHER... OR..." and they're summed. They don't mean "... AND ..." because they are not multiplied. Addition of (mutually orthogonal) wave functions means "(exclusive) or" for the possible further developments which is exactly why the wave function never has to prepare for both kinds of measurements. We know that if one measurement will be done, the other won't be, and vice versa! The form of the wave function guarantees that.
The part of the wave function for Alice corresponding to "Yes" (marriage) will continue to evolev all the degrees of freedom including the spacetime geometry according to Bob's template, a black hole exterior in which the black hole is remembered in \(BH\), while the "No" part (single infalling Alice) will evolve all the degrees of freedom to match the spacetime with the black hole interior. The same initial quantum amplitudes will be evolved in two different ways. One may consider both types of states but they're added at the level of the wave function and it means "OR", not "AND", so one doesn't discusses the degrees of freedom in \(A\) and \(R'\) simultaneously.
There exists no observer who could measure correlations between \(A\) and \(R'\), i.e. the interior and the late Hawking radiation, which is why the Hilbert space \(\HH_{A}\otimes \HH_{R'}\) doesn't appear in the physical description of anything by any observer. If Alice finds out she is inside the black hole, she will evolve her wave function in such a way that she keeps on using the degrees of freedom in \(A\). For Bob, or Alice who married Bob, the same degrees of freedom will be unitarily evolved into some degrees of freedom in \(R'\). There is no contradiction here. The reference frames of Bob (later in his life) and (single) Alice differ by a huge boost, \(\exp(i M J_{rt})\) where \(M\to\infty\), which represents the jets that Bob had to use to escape from the vicinity of the horizon, and this operator – in combination with the complicated "thermalizing" ordinary evolution in time – actually reshuffles the degrees of freedom in quantum gravity so that \(A\) is mapped to a subset of \(R'\).
I think that AMPS, Harlow, Bousso, and everyone who has "okayed" the AMPS argument as of today is just denying *everything* about complementarity. They couldn't have ever believed it exactly because they think that quantum mechanics must be prepared to answer combinations of measurements even though these combinations can't actually occur simultaneously.
It's pretty much exactly the same error that the people who use the classical intuition make when they try to understand quantum eraser – or even the double slit experiment whose careful analysis contains all the wisdom about quantum mechanics, as Richard Feynman has said.
A beginner may ask which slit the particle went through – it's a good observable – or what is the periodicity (distance between adjacent interference maxima) of the produced interference pattern. However, an important point is that if we measure the "which slit" information, there won't be any interference pattern, so we won't be able to measure its periodicity. So quantum mechanics doesn't have to answer – and doesn't answer – the questions about "which slit" and "what periodicity" at the same moment. We either observe the particle-like properties, or the wave-like properties of a particle. It's impossible to do both.
The case of the observer complementarity is analogous. In fact, it is a special example of the same complementarity. The complementarity in the double slit experiment boils down to the uncertainty relationship commutator \([x,p]=i\hbar\) for the observable \(x\) knowing about "which slit" and \(p\) knowing about the "wave-like properties". If you measure one, you disturb the physical system so that the measurements of the other will be different than if you hadn't measured the first observable. Effectively, the wave function only depends on \(x\) or only on \(p\).
It's the same thing with the regions \(A\) and \(R'\) in the black hole complementarity. If you measure an observable (a qubit) in \(A\), it means that you have perturbed your system in such a way that the spaceship fell inside the black hole and you're doomed. This event guarantees that you won't be able to measure qubits in \(R'\) so there can't be any contradiction. Effectively, the wave functional may be expressed as a functional of variables including the field modes in \(A\), or those in \(R'\), but not both of them. Just like \(x\) and \(p\), these field modes don't exactly commute with each other.
I could mention many experiments about "the foundations of quantum mechanics" – well, several of them have been discussed on this blog, like the delayed choice quantum eraser, in which many people are doing the same general mistake. They think that quantum mechanics is obliged to simultaneously answer questions even if the answers to these questions can't be simultaneously measured. But quantum mechanics isn't obliged to do this thing – even though this thing is "automatic" in any classical physics theory. Quantum mechanics only predicts probabilities of actual outcomes of actual measurements. And the probability of combined outcomes that can't appear simultaneously is just zero.
Yes, I have had the same feeling many, many times. Some people around you seem to know and understand everything, use the terms "quantum mechanics", "complementarity", "string theory", and many much more advanced features of it all the time. But a critical event arrives and those people reveal their new opinions which apparently prove that they couldn't have ever understood the essence of these concepts and they may have been just parroting some other people who either understood the concepts or they didn't and they were parroting someone else, and so on. ;-)
Finally, we seem to be converting in state where it's clear that no one really understands why string theory is right and the only possible theory of quantum gravity, how quantum mechanics and complementarity actually works, and so on. The questions that people actually misunderstand seem to be increasingly elementary, dating back to an increasingly distant past.
Wave function isn't objective
There's another interpretation of the main error behind the firewall paradox argument – one that's been discussed many times on this blog, too. Many people – apparently including AMPS, Bousso, and others – think that the wave function is an "objective observable" that must have the same value for all observers.
But the wave function isn't a classical observable, which is how such an object should be called. The wave function is a conglomerate of complex numbers ready for a subjective observer to compute probabilities.
Whenever observers may share their measurement and perceptions, both classical physics and quantum mechanics guarantee that there won't be any material contradiction between them. Classical physics guarantees that in a simple way, namely by accepting that there really exists an objective reality and all the observers just reflect it. That's a simple way to avoid contradictions between different observers in contact and many people think it must be right.
But it is not right. Quantum mechanics also guarantees that the contradictions are avoided but it achieves this goal without assuming an objective reality. The absence of contradictions requires a proof that is in principle simple but requires some mathematical abstraction and symbols. The objectivity of the reality is no longer a philosophical dogma in physics – in fact, it's wrong.
In this black hole case, Alice will be able to measure quantities in \(A\), a region inside the black hole. Bob will be able to measure qubits of the late radiation \(R'\). Complementarity in the old sense says that \(A\) and \(R'\) observables are analogous to \(x\) and \(p\) in basic quantum mechanics: they don't quite commute with each other. So one shouldn't be able to measure them simultaneously, without disturbing the other measurement.
The "without disturbing" comment is irrelevant here because \(A\) and \(R'\) may be spatially separated so no signals can propagate in between them. Nevertheless, it is still true in quantum gravity that one can't measure \(A\) and \(R'\) simultaneously. This almost sounds like a game with the English language but the word one is completely essential here. The problem with the attempt to measure them simultaneously is that Alice and Bob aren't one; they are two. And it makes a difference.
Because Alice and Bob won't be able to share the results of their measurements, they aren't one and it is actually totally OK if these two people measure, at the same space-like slice i.e. simultaneously and without a mutual influence, observables that don't commute with each other! You may imagine that Alice measures \(x\) of "a particle" (just a representation of a degree of freedom) while Bob measures \(p\) of the "same particle" even though the operators don't commute with each other. There's nothing wrong about it because Alice and Bob won't be able to compare the results.
As I wrote above, Alice has a wave function that contains terms in which she decides to stay out of the black hole; and terms in which she falls into the black hole. These are orthogonal terms in a wave function, mutually excluding possibilities, so they can't occur at the same moment. That's why it's OK if one of these two possibilities allows Alice to measure something she calls \(x\) and the other one allows Alice to measure something she calls \(p\) – and it may indeed correspond to "the" \(p\) describing the complementary degree of freedom to \(x\). What she will actually be able to measure depends on her decision about Bob's marriage proposal at \(t=0\). But this dependence is no more mysterious than the dependence of the ability of an experimenter to measure \(j_x\) or \(j_z\) of an electron according to a button he pressed to change the orientation of his detector.
The many-worlds interpretation is unphysical for many reasons but let me use some MWI jargon here. In the language of MWI, the measurements of \(A\) by Alice and of \(R'\) by Bob occur in "different parallel universes", however paradoxical it may sound. MWI doesn't provide us with any internally consistent recipes when the worlds should split (this is due to MWI champions' fantasy and random and mostly incorrect prejudices) but if you did MWI right, you would have to guarantee that Alice and Bob are in different parallel Universes that will no longer merge again, and they may therefore differ in many aspects that would have to be "single-valued" in a "unique objective classical world".
So at the end, I think that Polchinski and everyone else is still doing the basic anti-quantum mistake here, namely assuming that even for Alice, the result of measurements done with \(R'\) are "objective facts", and for Bob, measurements done by Alice are "objective facts". But they're not objective facts. Quantum mechanics shows that objective reality is just an emergent phenomenon and the event horizon is a place where a part of the objective reality gets immersed (the opposite of "emerge").
And that's the memo.
Alice moves along the yellow world line, Bob along the orange one. At the intersection of the yellow and orange world lines, Bob proposes to Alice. She has some probability amplitudes to say Yes and No, respectively. The "No" part of the wave function – which occurred on the picture above – leads to an evolution that reinterprets all the degrees of freedom in her Universe according to the infalling observer's template, including degrees of freedom that are interpreted as being localized in \(A'\). The "Yes" part of the wave function corresponds to the evolution that wasn't captured by the picture above: Alice marries Bob and continues with him on his orange trajectory through the (only approximately objective) spacetime. All their degrees of freedom would evolve into degrees of freedom interpreted as perturbations localized outside the black hole, which is the case of Bob even if Alice tells him "No". A point is that for us, observers outside black holes, the outcome of detailed measurements inside a black hole are not only impossible to be found; they're unphysical even in principle. The same is true for infalling observers' musings about the distant infinity and the Hawking radiation in that faraway realm that they will never see shining again. Similar questions about the events on the other side of the "Iron Curtain" are on par with questions "What would have we measured if we decided to measure \(j_x\) rather than \(j_z\)?" or "What would have happened if Hitler avoided a mistake and won a war?". Such questions can't be answered because the subsequent evolution would depend on lots of other/new decisions of Nature's random generator that can't be uniquely determined and that aren't a subset of the "set of facts" in the world where we live.
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