141. On information loss paradox, statistical and quantum mechanics
ASTRO, COND-MAT, HEP-TH/PH — By Dmitry Podolsky on December 15, 2008 at 4:54 pm
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Recently, I got into the discussion of information loss paradox in spacetimes with timelike and spacelike horizons (that is, black holes, de Sitter and staff like them). Let me remind you what is the issue (see for example Susskind’s recent book for details).
Suppose you prepare some quantum state descrbing a given physical object. This presumes you know the Hamiltonian describing your object and the corresponding Hilbert space of states; then you construct some linear combination of the basis vectors in this Hilbert space – coefficients in the linear combination are complex numbers.
Then, the physical object falls into a black hole, and the latter evaporates through Hawking radiation. Initially, you had a lot of information about your system – you knew all the coefficients (that is, their absolute values + phases) in the linear combination of basis vectors describing the initial state. As it seems, all you see at infinity is the bath of quanta of Hawking radiation with random quantum phases and a well known (thermal!) spectrum. So, was the information about the initial state lost or not during the process of black hole evaporation?
I think that the information is not lost – instead, I would expect tiny deviations from the thermal law in the spectrum of the Hawking radiation to exist. Why would I? The reason is that I know several things about de Sitter space (which looks somewhat like a black hole 
in the static patch – there is static horizon and thermal Hawking-like radiation from it that observer living in a corresponding Hubble patch registers).
First thing to note is that superhorizon physics does look like classical in de Sitter (foe example, you could recall the quasi-classical picture of eternal inflation I discussed on this blog so many times), but only if we neglect the exponentially decaying modes. If we do not neglect them and are somehow able to measure their amplitudes and phases, information about the quantum state of a scalar field in de Sitter space remains preserved. The problem is that it is extremely hard for us to measure an exponentially decaying mode, especially taking into account that we only have finite Hubble volume to make measurements in. I would expect some similar mechanism of information preservation in black holes.
But let us forget for a moment about de Sitter and black holes. Doesn’t a similar problem exist already at the level of simple quantum mechanics, without turning effects of gravity on?
Quantum mechanically, all of us are described by density matrices that know in principle about every single electron in our bodies. And still – the world around us is almost perfectly classical. Is there an information loss about, say, quantum phases of basis states our density matrix can be decomposed into? Clearly, the quantum mechanics is Hermitean – while the system evolves in time (my bus goes from Espoo to the University campus in Kumpula), the information about all electrons and nuclei – building blocks of our bodies - remains preserved.
The puzzle resolution, as we know it from the quantum mechanics, is decoherence. My density matrix tends to acquire a diagonal form, while off-diagonal elements become small exponentially quickly due to interactions of our subsystem with other subsystems – with armchair subsystem, keyboard subsystem and monitor subsystem in my particular case
Still, I do not feel I am exactly happy about the decoherence argument. Consider a density matrix
(1)
with at least one 
and 
non-zero. If we choose
(2)
where
, (3)
then
. (4)
This thing oscillates and does not approach any interesting time independenent asymptotics, which we could interpret as a decoherence limit (i.e., the limit of classical physics).
How to make our statement about decoherence physics somewhat more precise? And at this point here I find a recent PRL paper by Peter Reimann called “Foundation of statistical mechanics under experimentally realistic conditions“, which make things somewhat clearer to me (most probably all this is well understood by experts and I am just arrogant little … but I’ll continue 
).
He says – let us make two assumptions. The first one will be related to the quantity
. (5)
Since the number of energy levels N in a realistic system is tremendous, 
is extremely small (it basically behaves as 
).
Next, let us focus on a part of the full Hilbert space – only vectors 
such that 
(clearly, stadying how quantum system becomes classical, we are interested only in this part of the Hilbert space).
The second assumption is that our experimental device with which we measure the properties of the quantum state has a finite range of possible outcomes of measurement within this part of the Hilbert space 
.
Then, he says, let us consider an operator
(it clearly corresponds to quasiclassical, diagonal density matrix) and focus on the behavior of mean square deviation
After some relatively quick algebra it turns out that
, (6)
and since is so small for microscopic systems, the deviation from the diagonal density matrix is also extremely small.
The physical conclusion, I guess, that comes from (6) is that under experimentally realistic conditions it is impossible to control the values of for all those extremely dense energy levels of the spectrum of a given system (apart from the fact that the energy levels are “mainly concentrated” within some interval).
Would it be possible to derive an analogue of the inequality (6) for de Sitter space (using the coarse graining at Hubble scale presumably – since we only can measure the states with 
) or BH? I was thinking about it a bit but don’t immediately see how one could do that…
20 Comments
Dear Dmitry,
great that you’re brave enough to reveal your belief in information preservation and more importantly, you write more advanced things about it, too.
Let me now pretend that we’re isolated from the distraction of those who disagree with these basic, pretty established points.
I still see some additional question marks in the understanding of the situation by one of us or both.
You’re talking about some exponentially decaying modes that are supposed to preserve the information from the black holes. Well, I feel that you must have wanted to say something more refined than that.
What I mean is that if you have two states, psi+exp(-t)alpha and psi+exp(-t)beta, they just converge to the same state psi for infinite “t”, and the information is manifestly lost, isn’t it? (You may imagine density matrices instead of psi, alpha, beta.)
The problem is that even if the t=0 vectors psi+alpha and psi+beta were orthogonal, the “infinite t” limits are guaranteed not to be orthogonal but collinear, so this is not how unitary evolution can look like. So a linear deviation from a predetermined state that decays exponentially is not able to store the quantum information, I think. It’s the most typical representation of a continuous gradual loss of information.
It’s very essential that you have “truly different” states for large “t”, in the sense of their distance in the Hilbert space, I think.
What may be exponentially decreasing is the “effective” difference of the pure state from the density matrix, measured at a certain resolution. Still, the thermal density matrix has a large coarse-grained entropy while pure states have -rho.log(rho)=0. They can’t be the same things but it might be enough to trace over some “very fast modes only” to get the mixed thermal density matrix from the pure one, and the number of DOF that you need to trace over might be dropping to zero. That could be the sense in which the difference decays exponentially.
A task for you: do you think that if you use the Fock space basis for the thermal radiation, the information could be preserve purely in the phases of the different coefficients, but the absolute values could be universal? This sounds somewhat plausible (and might be close to the truth for generic black hole microstates) but in this strict form, I think that it is ruled out. Why? Because any localized state with a given mass can be considered a “very special case” of a black hole microstate (BH with the same mass), and for truly non-BH microstates, it seems clear that even the amplitudes in the radiation change.
The sentence “all of us are described by density matrices” could be confusing or misleading, too. Well, density matrices are just a tool to deal with incomplete information. It is always possible to imagine that “in principle”, we could know our pure states. The density matrix is a mixture of pure states that only describes the same kind of ignorance that you could have in classical physics, too – described by the probabilistic distributions on the phase space. The “deepest” theory doesn’t have to have these distributions, it’s just a math trick to deal with our ignorance.
Moreover, density matrices are only useful if you actually start to trace over some irrelevant degrees of freedom, and then it is no longer the case that the evolution must preserve the entropy – like from the exact Hermitean Hamiltonian evolution. That’s what you do in the section about decoherence, too.
Finally, when you talk about decoherence, it’s great, but isn’t it exactly another textbook example of a model where the information does get lost? Decoherence means de-coherence, i.e. the forgetting of the relative phases, which is a kind of loss of information. You can only derive it if you trace over some degrees of freedom.
If you trace over all degrees of freedom except for low-energy field theory modes (with frequency close to the temperature), do you think that the (exact, unitary) black hole radiation will look exactly thermal for each microstate? If it is the case, you can interpret the “forgetting of the information” as a case of decoherence. I would like to see a more detailed calculation what you exactly need to trace over in order to get the thermal radiation.
It seems to me that tracing over very-high-energy modes (as measured at infinity) can’t be enough because these modes don’t carry almost any particles (the Maxwell-Boltzmann decrease at high energies), so the tracing over them doesn’t make much difference to the density matrix. Is this intuition misled? But how do you identify the subset of DOF that you must trace over in order to restore the thermal character of the radiation? Maybe, you need to trace over a “1% of the low-energy modes”, too. It might be that for a large black hole, it is enough to trace over a very small percentage of the low-energy modes carrying most of the Hawking particles, and the remaining density matrix will be almost exactly thermal. The number 1% could decrease with the BH size in Planck units.
Is that possible that this is how you can get the thermal matrix from the pure state density matrix? It may be that this question is equivalent to some questions answered in the recent Susskind paper about fast scrambling. Curious about your answers.
Best wishes
Lubos
Dear Lubos
Nop, I think, you can measure psi-psi correlator, which will always contain an exponential tail. There are associated effects of interference between the modes etc. etc. Information does get lost, when you introduce coarse graining – i.e., when you admit you are unable to measure the exponential tail
In reality, there is no such thing as infinite t limit, all your measurements are performed at finite t (your measurement device does not exist forever, neither you do
), and the states remain orthogonal at any finite t, as you said.
I would expect that for BH even absolute values are not completelely universal, but forgetting about BH and trying to answer the abstract question – I think, yes.
Let me now go completely crazy and say the following thing – suppose that correlation functions a_k a_{-k} are not zero (yes, that is, corr. functions including two annihilation operators or two creation operators).
In a flat spacetime and a time independent problem, only a_k a_k^\dagger are non zero. However, for a curved spacetime or time-dependent problem the anomalous correlators above can also be non-zero (their physical meaning is the spontaneous amplitude of pair creation from vacuum). Now, when you said that the state is thermal, you actually said that a_k a_k^\dagger are given by the Boltzmann distribution, no more than that. Then, I can say that I also have anomalous correlators that carry information about quantum phases of the modes, and these correlators can in principle encode quite a bit.
Yes, if we also know the quantum state of the environment, which we don’t. Essentially, what I wanted in the post to say is that the evolution of the density matrix is unitary.
Again, you need to introduce coarse-graining in order to get a decoherence. Also, if you are able to measure (do not ask me how) a quantum state by a quantum device, there is no information loss (that’s what colliding QFT quanta do all the time – interacting with each other, “measuring” each other); but if you measure a quantum state by a classical instrument, then you have decoherence, wave package reduction and all that.
True, but information may be also stored in the phases. Take (quasi) de Sitter for example. CMB fluctuations come from the vacuum fluctuations, what you see on the sky is the phases of different modes.
Cheers,
Dmitry.
There is a third option, perhaps information is not lost because Hawking Radiation does not exist.
“black holes do not radiate” [1]
“The possibility that non-radiating ‘mini’ black holes exist should be taken seriously; such holes could be part of the dark matter in the Universe” [2]
“the effect [Hawking Radiation] does not exist.” [3]
“2) infinitely delayed Hawking radiation; 3) infinitely weak chargedness of black holes” [4]
“it is possible that the behavior of the black hole is stable” [5]
[1] arxiv.org/abs/gr-qc/0008016, Trans-Plankian Modes, Back-Reaction, and the Hawking Process, Prof. Dr. Adam D. Helfer (2000)
[2] arxiv.org/abs/gr-qc/0304042v1, Do black holes radiate? Prof. Dr. Adam D. Helfer (2003)
[3] arxiv.org/abs/gr-qc/0607137, On the existence of black hole evaporation yet again, Prof. VA Belinski Paper. (2006)
[4] http://www.wissensnavigator.co.....CKHOLE.pdf Abraham-Solution to Schwarzschild Metric Implies That CERN Miniblack Holes Pose a Planetary Risk, Prof. Dr. Otto Ressler (2008)
[5] arxiv.org/abs/0808.2631 On the Stability of Black Holes at the LHC, M. D. Maia, E. M. Monte (2008)
Dear TJ Tankers
I think the following site might seem interesting for you:
LHC live cams
Cheers,
Dmitry.
I saw that web cam once before. Well done!
Dear Dmitry, it seems that there is some disagreement that goes beyond terminology here. When you argue that an exponential decrease of the psi-difference is not (…) a loss of information, you talk about the psi-psi correlator.
What does it mean? You have TeX here, could you please write it with brackets?
What I meant by psi was the wave function and wave functions don’t have any “correlators” (since they’re not “operators”). A wave function converging to a universal “psi” – or a density matrix converging to a universal “rho” – are textbook examples of processes where the information is lost.
You can’t measure any “exponentially decreasing deviation” from this psi in quantum mechanics. A basic feature of quantum mechanics is that what used to look like very small (but in principle measurable) effects in classical physics is replaced by finite (quantized) but low-probability (i.e. almost certainly unmeasurable) effects in quantum mechanics. Having 0.999999 psi + 0.001 alpha means that you have a 99.9999% probability of measuring exactly psi, as classified by measurements of any complete commuting set of observables. In other words, you have a 99.9999% change of proving that the state is identical as it was for another initial microstate. The information is lost.
This is exactly the kind of evolution that must NOT happen if the information is preserved.
Also, it’s hard to imagine how the finite lifetime of the black hole could save you. The exponential decrease in your tail is just fast and the black hole lifetime is very long. If you think that this qualitative guess is wrong, I want to see your parameteric Ansatze that change the result. When the final decay product is e.g. a stable extremal black hole, the lifetime is even infinite. Would you still expect the exponentially falling deviations here?
Concerning your a-a correlators, can’t you just try to construct the state in the Hilbert space (pure or mixed, whatever you need) that produces your hypothetical correlators? I feel that things like that can occur, but in some sense, it’s not clear to me why you’re postulating that the a-a correlators are changed while the a-adagger correlators are not. What’s the principle that preserves one but not the other? Note that a nonzero a-a correlator links the coefficients of the wavefunctions with different numbers of particles. And if you allow both types of coefficients to change, aren’t you talking about a completely general density matrix now?
I agree that in principle, the evolution of density matrices is unitary – but that’s only and exactly in the case you follow all (including environmental) degrees of freedom, a requirement that you seem to deny in the same sentence, confusing me a lot along the way. If you’re losing (averaging over) information in the environmental degrees of freedom, you introduce nonunitarity to the density matrix evolution: the density matrix at t1 is no longer conjugate to the matrix in t2. Decoherence is an extreme example of such a nonunitarity, and it’s almost anywhere where you have any environment.
The CMB mental image for the phases (to be used for the BH microstate’s radiation) could be helpful, tx. Doesn’t it imply a fix or refinement for your previous a-a correlator picture?
Dear Lubos
Sorry, cannot allow turn TeX in comments, it well break my server down
Psi means psi-operator
\Psi (t,r) = \sum_n a_k u_k (t,r).
The expectation value of \Psi\Psi^\dagger gives the number density in the point r at the time t. I want to discuss this thing instead of the wave function, since we are actually interested in QFT in BH background, not in quantum mechanics.
Fine, unfortunately these text books don’t explain the information loss paradox in BH, do they? (But don’t worry, I am not going to question the foundations of quantum mechanics
)
Yes, I can, if I have infinite time to make infinite number of experiments with exactly same initial quantum state.
The lifetime of BH does not play any role in my considerations. I would like to ask you the following question – suppose you created a pair, one particle went inside the horizon, another – left to infinity. The question is – were these two particles created in a coherent state, in other words, are their phases correlated?
I think, both aa and aa^\dagger are changing, you have two coupled differential equations for these correlators – one is kinetic equation for the number of particles in the outgoing flux, another – for the quantum phases.
Ok, of course, you are right. I should not have started talking about density matrix in the first place.
The picture is a kind of similar, I think. During inflation, modes cross the horizon and adiabaticity condition for them breaks down. Then, particles are created immediately after horizon crossing (in a coherent state, see above), after that modes freeze, as well as their quantum phases. After the end of inflation, the modes reenter the horizon, and we see the phase picture on the sky.
I think – one way to treat this physics is to write kinetic equation for the particle numbers in modes and an equation for the anomalous correlator coupled to the kinetic equation.
Let me now just shut up and write a paper – I got quite excited about this picture yesterday.
Cheers,
Dear Dmitry,
thanks but if you mean field operators by “Psi”, why don’t you talk about normal, actual, realistic, relativistic quantum fields such as A_mu or Phi (Klein-Gordon) or g_{mu nu} instead of a “Psi” that doesn’t really exist (with this name) in any realistic theory?
When you say “I am not going to question foundations of quantum mechanics”, you violated this pledge just a paragraph earlier. If you say that QFT is not a theory in quantum mechanics that has a wave function(al) “psi” (which is more relevant for encoding final states of evaporations than field operators, anyway), then you are surely questioning the foundations of quantum mechanics.
Also, I don’t think that you have given arguments that these questions should be studied by correlators of field operators and what these correlators should be for the microstates. Note that in the UV, semiclassically, the correlators are universal. Only long-distance correlators are slightly affected by the background. And if they depend on the microstate, non-thermally, you haven’t defined a rule how they should depend and what they should share.
When you’re talking about having infinitely many copies of the same initial states to determine whether the information is lost, you are redefining what the information loss means.
Wave functions or density matrices that converge to a universal limit surely would mean that the information would be lost. What “more obvious” or “more serious” information loss can you possibly think of as long as quantities are continuous functions of time?
Of course that if you imagine that you can measure the “wave function itself” and “arbitrarily exactly”, then no information can ever be lost. For example, if a book falls into a black hole, it creates some characteristic gravitational fields that are imprinted into the ringing modes of the black holes which surely leave traces in the wave function. These traces decay exponentially, with the quasinormal modes, but according to your rules, you can magnify them as much as you want, so they will never disappear.
But that’s not what we mean by information preservation in quantum mechanics. Information in quantum mechanics is something that can be measured in qubits, and the whole “exponentially tiny” infinitesimal neighborhood of the vector “psi” in the Hilbert space has information 0 qubits in the limit when the neighborhood is small. Just calculate it. Make a mixed state out of all states of the form psi+exp(-t).somealpha where somealpha is some other vector and t is large. And calculate -Tr(rho.ln rho). Clearly, the entropy is zero as t goes to infinity.
You indeed seem to suggest that the wave function or density matrix is an “observable”. But it’s not. Saying that the wave function is an observable that can itself be measure is a brutal violation of basic and general rules of quantum mechanics, and these rules are essentially for every QM theory including every QFT. Sorry, you really seem to misunderstand something basic here.
The information is there only if there are distinct states in the Hilbert space that generate an inherently multi-dimensional (hugely multi-dimensional) subspace of the Hilbert space – of dimension exp(entropy) or so. With your redefinition of the preserved information, everything goes and the “preservation” would be a meaningless tautology. The problem you’re trying to solve is not the information loss paradox. It is not even a problem because the “problematic” situation cannot occur because of very rules of logic.
I can’t believe you don’t know what I am saying. But I am still curious about your prescriptions how the a-a and a-adagger correlators are supposed to depend on the microstate even though I haven’t seen any evidence so far that you actually have some sketch of something that makes sense here.
Best
Lubos
Dear Lubos
Ok, I see that you will not allow me to shut up
The sketch is the following. If you consider, say, Schwinger particle production in a strong electric field (or generation of inflationary perturbations), you will find that pairs are created in a coherent state, that is, phases of particles and antiparticles are correlated. If you measure a spin projection of a particle to the z axis, then you have EPR effect, and the spin of antiparticle is also getting fixed.
In fact, pairs are always created in a coherent state for any particle production process in a strong external field. I bet that particles and antiparticles are also created in a coherent state in Hawking radiation. So, the question is – what happens to particle behind the BH horizon if you measure a spin projection of antiparticle at infinity?
Cheers,
Dmitry.
Hi again Lubos
Concerning your other questions:
That’s because Psi are not field operators, a field operator would be
Phi = Psi + Psi^\dagger,
I want to catch information about anomalous correlators < a_k a_{-k} > as well. Psi (as I defined it above) is introduced in Landau-Lifshits, Quantum mechanics, chapter devoted to Fermi-Dirac and Bose-Einstein statistics and used in many other books on QFT methods in condensed matter physics, for example, Abrikosov-Gor’kov-Dzyaloshinksii.
Not sure about that. As I said, pairs of quanta of Hawking radiation are probably created in a coherent state, and field operators (in secondary quantization) nicely capture this fact, while it would be hard to see this in first quantization.
You know, what I would actually like to see is the answer to the following question. Take some mixed state in the form you proposed. What is the time t_* such that entanglement entropy is zero for any t>t_*? Does t_* depend on the form of initial mixed state?
Not sure where this came from. I was talking about observables of the form which give the number density in a given point of space. Sure, this number density may become of order of 1, but it will still give you a probability to find a particle in a given point of space, and this probability may be extremely tiny, but still not zero.
Cheers,
Dmitry.
Dear Dmitry,
concerning the first new comment of yours. Sure, complete agreement, the particle-antiparticle pairs are created “entangled”. But because of black hole complementarity, this is another tautology because the black hole interior is just encoding some degrees of freedom from outside in a very complicated way.
The correlation between spins and other properties of particular excitations of this form (inside vs outside) is just the simplest example of black hole complementarity. Denying this correlation inevitably leads to the quantum xerox paradox.
Yes, as you suggest, when a particle at infinity is measured to have a certain spin, it modifies the probabilistic amplitude for measurements inside, too. So there will be correlations like in any EPR measurement.
But in this setup, they will be less interesting, not more interesting, than in generic normal EPR setups, because of roughly two reasons:
1) More complicated measurements further from the horizon will be terribly, chaotically scrambled. That will “effectively” mean that the low-energy measurements inside and outside are independent. For example, the result of a measurement of J inside the black hole could be some other quantity K outside the black hole, read from the 10th decimal point.
And in reality, these links are even more complicated. But even in the simple example, if you measure J to be 8.9793238, many people won’t even recognize that it came from K=pi outside 
. When the code is more complicated and not fully known, the correlation will be unfindable from broad, qualitative features of the approximately measured numbers.
2) The observers inside the hole are causally disconnected from those outside, so they will never be able to compare their experiments and prove the “EPR” correlations, anyway.
Best wishes
Lubos
Dear Lubos
Just the opposite, this argument actually makes sense of black hole complementarity and quantum xerox paradox, which are linguistic exercises without it
Using complementarity try to answer to the following questions quantitatively:
a) are all degrees of freedom doubled or some of them – not? (ehrrrmmm, what is the entropy of BH and what is the entropy associated with degrees of freedom outside it in the remaining not so small part of the Universe? what’s about other BHs than this particular one – is there doubling of all the information for them, too?)
b) what is the actual dynamics of quantum phase of the state while antiparticle goes inside the horizon, while the particle – outside? How the horizon affects this dynamics? Surely, there should be some quantitative effect of horizon’s presence in the correlator of phases.
If you wait for a bit (or do you prefer to switch to email conversations? I do not want to publish it on the blog yet), I will actually construct the formalism for quantitative treatment of these questions.
Further, agree with both your points. However, I think, these are secondary questions. What we wanted to figure out originally is whether the information (or at least some information
) is preserved in principle.
Cheers,
Dmitry.
Hi Dmitry,
concerning the second comment, you can define “Psi” to be the annihilation part of a field operator. But such a “Psi” is not a local field, it doesn’t (anti)commute with the same “Psi” at spatially isolated points.
Consequently, superluminal correlations and signals and evolutions as measured by this “Psi” can occur, and it is impossible to prove the semiclassical information loss using this “Psi” only – because in the language that doesn’t use the full Psi+Psidagger operators, there’s no causality constraint.
Concerning t*, if you calculate the decoherence, the time is negligible, even if the objects and interactions are relatively small and weak. For black holes, as long as you treat anything like the environment, the decoherence is the fastest you can have, and t* would be much (exponentially) shorter than the Planck time. If the information could only be preserved in this way, it would be effectively lost instantly.
In low-energy modes that you don’t integrate out, the information is disappearing exponentially with the lifetime given by the low-lying quasinormal modes (essentially close to the black hole radius). All other more detailed processes killing the information are much faster.
In your last paragraph, you once again confirm that you are making the undergraduate mistake of treating wave functions – and probability distributions – as observables. They’re simply not observables. What is observable is the actual value of a formerly classical degree of freedom such as “x” or “p”. But “Prob” is not an observable, and if the probability that you measure something different given microstate “A” differs from the probability that you measure the same thing given microstate “B” is exponentially small, like exp(-entropy), it means that the information has been lost.
That’s how information is normally getting lost in the black hole that is getting stabilized, by the ringing modes and gravity waves radiation that bring the hole into the standard static shape.
Again, your idea that the information is not lost because the initial state is still encoded there, with an exp(-t/t0) coefficient, is just completely wrong. If deviations of states from a “universal limit” – or deviations of correlators of operators in these states from a universal (thermal) limit – universally decrease exponentially with time, it means that the information was getting lost. The only way to explicitly show that the information didn’t get lost is to actually find correlators (of some “harder” quantities) that do depend on the microstate and don’t decay exponentially.
Best
Lubos
And hi again
Sure, it principle, acausality can be there. However, you know that \Phi\Phi^\dagger is trivially related to the number of particles, while the anomalous correlator \Phi\Phi – with the correlation of phases. And surely both quantities behave causally (although the kinetic equation for n_k is integro-differential, that’s true).
Ok, I am sure we understand each other, exponential tails is a dead end.
?? Did you want to say that condmat experimentalists who measure the current through the SNS mesoscopic contact (with the precision up to one electron) actually measure only Santa knows what? (0 component of the 4-current is my \Phi\Phi^\dagger)
What if instead of your picture related to microstates one uses Heisenberg operator picture? Should get the same results.
Cheers
Dmitry.
Hi Dmitry,
I agree that if someone doesn’t realize that entanglement links a priori different degrees of freedom and makes them correlated, he or she must view the word “complementarity” as a piece of linguistics. Well, it might be true that I wasn’t an element of this set.
Nevertheless, complementarity is not fully *explained* by saying “entanglement” because the evolution in semiclassical gravity doesn’t provide us with a mechanism that would be linking the pairs of degrees of freedom at slices before the formation of a black hole becomes obvious.
Note that in the Penrose diagram of a black hole, you may draw purely spatial slices that contain both points in the interior and the exterior, but these slices are a priori indistinguishable from normal spatial slices through a flat space.
Concerning your “quantitative” questions, I am not getting them. The first question is qualitative, so I am not sure how an answer might be quantitative, and I moreover disagree with the answer that you want to hear. More concretely, I don’t know why you’re saying that “all degrees of freedom are doubled”.
Quite on the contrary, the point of complementarity that helps us avoid the xerox machine paradox is that *none* of the degrees of freedom are doubled in the exact quantum description of the system.
On the other hand, if you take a semiclassical description where the linkage between the degrees of freedom is not manifest, the question makes no sense because there exists no link between the interior and exterior degrees of freedom – they’re independent at this level – so you can’t say that any of them is doubled. The closest thing you can ask is whether there are two regions of a black hole – outside and inside – and the answer is Yes.
But that doesn’t mean that the degrees of freedom are “doubled’. They’re two sets of degrees of freedom that are independent at the semiclassical level, but that are linked at the exact quantum level.
I am actually working on a variation of your question b), in some sense.
Hi Dmitry, I lost 20 minutes of writing a comment because my PC froze and I don’t have enough energy right now to redo it. Sorry. Merry Xmas, Lubos
Dear Lubos
I think I got your comment although your PC froze (is it the one that’s above or you actually have said more?).
Indeed, when I was writing the question (a), my neurons seriously misfired, and I just formulated the basic quantum xerox statement – what I wanted to ask is: are all degrees of freedom inside and outside BH linked? If not, then which ones are linked and which ones are not? What if your have several BHs – are all degrees of freedom outside them linked to degrees of freedom inside every BH? I may be wrong, but I have an impression that as long as you don’t take rescattering of Hawking radiation into account, both quantum phases of modes (of a massless scalar field for example) and associated occupation numbers inside and outside the horizon are correlated. But after a single act of rescattering (or interaction of Hawking radiation with matter outside the black hole) correlation is lost (quantum phases are rapidly randomized, occupation numbers may change due to particle production in the act of rescattering).
Thank you so much for visiting and sharing your thoughts with me. Merry Christmas for you and your family!
Dmitry.
Merry Christmas to you and your family!
Dmitry.
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