**The state dependence is a simple example of effective superselection sectors; Joe's firewall confusion is linked to his sleeping beauty mistakes**

I have finally found some time to watch more videos from Strings 2014. You may download them from the conference web page about talks or watch many/most/all of them on the GraduatePhysics YouTube channel.

The talks by Joe Polchinski, Kyriakos Papadodimas, and Suvrat Raju are among those that talk about the black hole interior. This blog's fans want to see Suvrat's talk, 25:40-25:50. ;-)

What Suvrat and Kyriakos say makes sense. I looked at Joe Polchinski's dissatisfaction, e.g. in his slides (PDF). Pages 20-24 and some of the following ones are dedicated to Joe's objections against Suvrat's and Kyriakos' picture. I find the causes of Joe's apparent unhappiness strange. He believes that Suvrat and Kyriakos ("PR") violate some general rules of quantum mechanics but all the contradictions quoted by Joe actually arise because Joe, and not PR, violates some laws of quantum mechanics.

Also, it seems to me that the same mistake that leads Joe to wrong answers about the sleeping beauty problem are not only behind his anthropic blunders but they have led him to the firewall dead end, too.

OK. Let me go to page 20 of the slides. Joe's very first equation is\[

\ket{\psi_{\rm vyp}} = \frac{

\ket{0}_B \ket{\psi_{\rm typ},0}_{B^*}

+ e^{-\beta\omega/2}\ket{1}_B \ket{\psi_{\rm typ},1}_{B^*}

}{Z^{1/2}}

\] where \(B^*\) is the complement to \(B\), we learn. The way how Joe phrases these things makes it rather clear that he always assumes that the degrees of freedom may always be split to two geographical regions, one region and its complement, which isn't really the case due to the black hole complementarity but I think it's not really the heart of Joe's anti-state-dependence sentiments so let me ignore these ultimately wrong operations and summarize the page 20 as Joe's version of the explanation why there's no firewall in the PR picture.

The "key issue" is mentioned on the following page 21.

Key issue: given a black hole in some state \(\ket\psi\), what reference state \(\ket{\psi_{\rm typ}}\) do we use? Is the state above but with the relative minus sign an excitation of the state above, or a typical state that should be classified as unexcited?The correct reply is, obviously, that the answer depends on the reference state. Unless we are constrained by something else, we may choose any reference state but the description in terms of low-energy degrees of freedom will only be applicable if the reference state is sufficiently similar to – roughly speaking, related by a relatively low-order polynomial of local field excitations to – the states that the system under consideration is actually found at.

As far as I can see, the very fact that Joe is asking the question proves that he hasn't even started to

*consider*the proposed claim that all these constructions are being built relatively to a reference state. If he had started to consider this proposed claim, he would have known that there couldn't be a unique answer to such questions. So page 21 is just a proof that Joe hasn't started to even

*consider*the proposed explanation.

On page 22, Joe reviews the PR construction of the interior operators in terms of the reference state but adds his own comment:

The issue is that when one specifies the reference state \(\ket{\psi(t)}(\psi)\), these become nonlinear operators \(P(n_A,\psi)\).It would be very bad for PR if they modified the Born rule but they do nothing of the sort. In fact, they never write anything such as Joe's function \(\ket{\psi(t)}(\psi)\) because it's wrong. The reference state that Joe calls \(\ket{\psi(t)}\) enters the PR formula for the interior operators but isn't supposed to opportunistically, exactly, and immediately vary depending on some particular state \(\ket\psi\) that is standing on the right side from the operator so that the operator acts on it. Instead, the vector \(\ket{\psi(t)}\) is kept fixed while we study the operators and measure their values etc.

This state-dependence is a modification of the Born rule, and is different from normal notions of background-dependence.

With a fixed reference state, the Hilbert space is completely standard, orthodox, and linear, the operators are also perfectly linear, they may be measured, and the probabilities of different values are calculated by the totally orthodox and standard Born rule. All of Joe's claims that the postulates of quantum mechanics are being modified are just pure rubbish.

The only thing that the state dependence means is that the Hilbert space with all the unexcited states, single-excitation states, more excited states, and all the operators the map these states onto each other are being built from scratch in a given physical context. The given physical context will allow us to add a finite – perhaps large but not too large – number of local field excitations but with too many excitations, the well-definedness of the local field operators is guaranteed to break down. It's not surprising at all. One simply can't create an unlimited amount of stuff into the black hole interior.

But whenever quantum mechanics is used to predict probabilities of some measurements, we must know how the operators act on the state that we have prepared. We must know what the operators that we measure are. If we don't know what the operators are, quantum mechanics obviously yields no predictions for the (unknown) operators' values.

At the end, the state dependence only means that two black hole microstates that differ by "more than a few" field theory excitations – microstates that are "far" in the PR sense – should be treated as two different superselection sectors. There isn't any canonical map between states in two superselection sectors even if the two sectors are mathematically similar. I can make this analogy very accurate and I can also explain why there can't be a global definition of local operators across superselection sectors in quantum field theory or string theory. Even if the two worlds look similar - like two macroscopically indistinguishable black hole states; or like two string vacua with a Standard Model at low energies – they are really two different worlds that can't be operationally obtained from each other, at least not by a simple action of a low-order polynomial of local operators. To say the least, if you defined a dictionary between the operators acting upon the two superselection sectors, the phases of all the creation operators \(a^\dagger(\vec k)\) may be redefined by a function of \(\vec k\). This turns one proposed convention to define the operators "upon all superselection sectors" into a different dictionary. The dictionary is clearly not unique.

Page 23 is just repeating trivialities about the Born rule with the wrong claim that PR violate the Born rule. Joe's mistake is simply that he varies the reference state all the time. But it isn't supposed to vary during the predictions of a single measurement at all. One is supposed to envision the relevant Hilbert space, depending on the initial conditions, and the operators that act on it. This relevant quantum mechanical model has everything it needs and the postulates of quantum mechanics are and have to be rigorously followed. The state dependence just means that for a given "region" of the black hole microstates, the operators have to be defined separately, from scratch.

Joe believes that either the operators have to be defined everywhere, across all the black hole microstates – across all the superselection sectors, if you use my discourse – or the regions have to be defined so finely that the Hilbert spaces are just one-dimensional and bizarre. But the truth is in between – none of these extremes are right. Quantum mechanics would be completely meaningless if the relevant Hilbert space were always one-dimensional. The relevant Hilbert space always has to be multi-dimensional for the operators not to be \(c\)-numbers and to be able to distinguish several results of measurements.

Page 24 brings nothing new. Joe still complains that the same states may be excited or unexcited, depending on the choice of the reference state. Indeed.

But it was page 25 that made me explode in laughter. Joe starts the page by asking two questions:

How to interpret \(\alpha \ket{\rm vac} + \beta\ket{\rm exc}\)?You would think that Joe is asking how to interpret two superpositions because he

How to interpret\[

\frac{ \ket{\rm vac}\ket{+z} + \ket{\rm exc}\ket{-z} }{\sqrt{2}}

\]

*doesn't know*how to interpret them. But the next sentence raises a

Problem: the interpretation is different if one writes it as \[LOL, that's hilarious. Joe doesn't know what the interpretation is but he's sure that the interpretation is different from the interpretation of the very same state written differently!

\frac{ (\ket{\rm vac}+\ket{\rm exc})\ket{+x}

+

(\ket{\rm vac}-\ket{\rm exc})\ket{-x}

}{ 2 }

\]

No, Joe, the interpretation of the same state is always the same. With a choice of the reference state, the experimentally accessible Hilbert space may be completely built. It's the same kind of Hilbert space as the Hilbert space in any quantum mechanical theory and it's used in the same way. One may identify some excitations as spin-\(z\) or spin-\(x\) electrons and one may add an apparatus to measure it as long as the apparatus is constructed from a sufficiently low number of excitations. If the apparatus can't be built in this way, it probably doesn't fit the black hole. The predictions for all the measurements of \(j_z\) or \(j_x\) and their correlations with the excitation level may be computed by the universal quantum mechanical formulae.

The state dependence is really just an expression of a tautology. In between two sectors of the black hole's Hilbert space that are significantly different – they don't differ by a simple enough polynomial of local fields – you can't travel by the action of local operators. So from the viewpoint of the local operators, these two sectors of the Hilbert space act as superselection sectors whose ground state and excited states should be thought of as different worlds, despite their macroscopic similarities.

Let me finally get to the sleeping beauty.

Joe Polchinski is a thirder and this simple error in the probability calculus has been explained to be the source of some of his wrong opinions about the need to have "typical stringy vacua" etc. But how can the thirder's fallacy explain the firewalls?

Now I think that these two errors share the same basic idea which is an incorrect, retroactive, acausal change of the rules of the game while we compute the probability. What do I mean?

If we use the notion of probability correctly, probabilities tell us how our knowledge about something is probably going to change once it changes. I know that I used the word "probably" so the "definition" is circular and it is not a real definition. But what I want to say is that the notion of probability assumes a "fixed situation" before we learn the new piece of information, and many things may happen after we learn the new piece of information, but these later events don't and they can't affect the probability that existed right before we learned the right answer.

According to Joe and other thirders, probabilities should behave differently. If something happens after the moment when we're supposed to learn the right answer, it should still retroactively modify the denominator that may define the probability in a frequentist way, among other things. Just because the sleeping beauty is woken up multiply times on the "heads" week implies that the probability of "heads" had to go up. Joe and other thirders actually believe that the probability of tails was already \(P=1/3\) on Sunday because the algorithm for the awakenings had already been decided.

But whether the sleeping beauty is going to be woken up twice is a

*consequence*of the result of the coin toss, so it just can't affect the probabilities of the different outcomes of the coin toss. So the beauty knows that \(P_{\rm tails}=1/2\) on Sunday night and because she doesn't learn anything that would change the odds of "heads vs tails" when she wakes up (both possibilities guarantee with 100% certainty that she wakes up at least once which is the only "evidence" she gets), the probabilities have to remain \(P_{\rm tails}=1/2\).

Joe's mistake in the case of the state dependence is analogous for the following reason. In the correct quantum mechanics, we must decide what the states in the Hilbert space – and operators – corresponding to different results of the measurements are, and we must keep this fixed. This choice is much less ambiguous in practice because we really evolve the fields according to the usual Heisenberg field equations.

But according to Joe, we are not obliged to deal with a fixed spectrum of possible outcomes of the experiment. The experiment may create them (and perhaps erase them) later. That's why Joe insists on changing the reference state "accurately" whenever some operator acts on anything. (This isn't really a well-defined rule because in a generic equation with many operators, there are many states, depending on which actions of the operators are included etc.) He thinks that this gives him as accurate answers as possible.

Instead, what it does is that it invalidates and erases any previous well-defined rules of the game – with known options (possible outcomes) whose probabilities used to be predictable. If he allows the evolution of the state vector to change what the question is, then there is no pre-existing question, and quantum mechanics cannot predict any probabilities! If he imagines that the post-measurement state \(\psi\) is the only state one could have gotten, and everything should suddenly adapt to this new state, even the state of the black hole before the measurement, then there was no uncertainty before the measurement and the measurement could have brought the observer no information. This wouldn't make any sense, of course. It is absolutely crucial that we are not changing the rules of the game – we are not changing the reference state – at the very moment of the measurement.

In some future, I plan to write a more technical post on the state dependence, the superselection sectors, and things like the holomorphic anomaly in topological string theory because there are many direct analogies (or equivalences, in the sense of one's being a special case of another) between the concepts.

(not so) off topic : maldacena's conjecture article has reached 10.000 citations today!

ReplyDeleteA cool milestone - I hope that Milner or someone will also send him a T-shirt or hoodie like Stack Exchange is sending me for crossing 100,000. ;-)

ReplyDeleteWow, the silence here is deafening. Apparently nobody competent enough to recognize your arguments for what they are -- obviously correct -- dares to speak out and mess with the Big Polchinski. Politics trumps science in academia, I guess.

ReplyDeleteThanks, Michael, but you exaggerate. The silence on the blog is because the blog readers are either non-experts or shy workers who are working on their research. I think that those that know (including Kyriakos and Suvrat) aren't really too afraid to disagree with Joe even though such fear does exist, too.

ReplyDeleteit seems to me that milner already gave him something like that a couple of years ago, some kind of coupon if i remember well.

ReplyDeleteNo, I know, but maybe it wasn't enough for Juan to buy an AdS/CFT-10,000 T-shirt. ;-)

ReplyDeleteOh my God, even though I had not yet time to read all of the sleeping-beuty articles in detail, agree that it seems that Joe P. has same bad misunderstandings abou basic probability theory (being a thirder).

ReplyDeleteAlso from teading about different methods to qualtise ST for example, it always seemed to me that as soon as one has defined everything, results of measurements can just be calculated and there is no need to continuously change the starting point of quantization.

Remember that anyone who might disagree with Lubos, like DN, has been banned, so only his sycophants remain.

ReplyDeleteRemember, anyone who disagrees with Lubos, or actually understands probability, like DN, is banned from the forum. So only Lubos's sycophants remain. So this comment has no chance of being accepted.

ReplyDeleteActually, Lubos, not everyone behaves like you, insulting those who disagree with them, and banning them. There is ample discussion of state-dependence in the scientific community.

ReplyDeleteSuppose that we change the SB puzzle, but only in the protocol for Tuesday after tails (T/t). The new protocol is that she is awakened, and, after a pause of a minute, is put back to sleep (but only on T/t). Again, she knows the full protocol in advance. In the first minute after awakening, she assigns equal probabilities for M/h, T/h, M/t, T/t. After a minute, if she is still awake, she excludes T/t and is left with equal probabilities for M/h, M/t, T/h. So in this case the 1/3 is correct. Do you agree with this result for the modified problem? If so, how can the changed protocol for T/t affect her relative weightings of the other three cases? What if she's only awake for a second?

ReplyDeleteOf course that I disagree with the result 1/3 here. After the "demo minute" or "demo second", her knowledge is exactly the same as in the original problem, so your problem becomes the original problem! And it's a way to formulate the right solution "by parts" which may be even clearer than all the solutions so far.

ReplyDeleteBefore she may be put to sleep, there are 2 possible coin states, heads and tails, and 2 possible days, Monday and Tuesday. By two Z2 symmetries, the probability of each of the 4 possibilities is 1/4. Would you really disagree with that?

Now, after the second or minute for the short awakening, the situation changes because she's getting a new information in general.

How much information she is getting by staying up depends on the coin state. If the coin shows "heads", she is getting no information because in the "heads" case, staying up is predicted for both days.

If one assumes that the coin is showing "tails", then she is definitely getting information by staying up. Before the momentum of "possible putting back to sleep", the probabilities of Monday-tails and Tuesday-tails were 25% each. That means that the conditional probabilities of Monday and Tuesday, assuming tails, were both 50%.

After the moment when she can be put back to sleep, the probabilities change. The "fluid" of the "Tuesday tails" just flows to "Monday tails" because Tuesday tails is getting riuled out. So the conditional probabilities of Monday and Tuesday, assuming tails, become 100% and 0%. They still share the same initial probability reserved for "tails" so this 100-0 translates to 50% - 0% in absolute probabilities.

The probabilities of heads-Monday and heads-Tuesday stay at 25% because if one/she assumes heads, one/she is learning nothing about the day of the week!

The result, after the minute/second expires, is of course the same 50-25-25 as before.

Dear Lubos, Thank you for your answer. Now suppose that instead of being put back to sleep, after a minute she is simply told, `it's T/t'. So in the other three cases, it is exactly the same as the previous modified protocol, she gains the same information after one minute.

ReplyDeleteSo, if she is told (which is 1/4 of the time), then it's 100% that it is T/t. If she is not told (3/4) then it's 50% that it is M/t and 25% each that it is M/h and T/h. This seems to give a total probability of .25*1 + .75*.5 = .625 that she is in one of the tails cases. The thirder calculation would give .25*1 + .75*.333 = .5.

What you wrote was meant to be

ReplyDeleteP(tails) = P(toldTT) P(tails|toldTT) + P(notToldTT) P(tails|notToldTT)

wasn't it? But this formula makes no sense because it doesn't express her subjective probability of "tails" at any single well-defined moment, neither by objective time nor by subjective time.

You're summing probabilities at "a" moment when she's told something with "a" moment when she's not told something, and these are not mutually exclusive because assuming "tails", she is both told (on Tuesday) and not told (on Monday).