Saturday, December 13, 2014

Entropy, temperature are not fixed linear operators

Similarly fields in GR. A simple demonstration of "state dependence" in quantum gravity

Kyriakos Papadodimas and Suvrat Raju have demonstrated that it's possible to embed operators describing the fields in the black hole interior into the Hilbert space of a black hole so that all the usual principles approximately hold.

Their construction doesn't imply that certain questions about the perceptions of the infalling observer have unambiguous answers – indeed, one may worry about the non-uniqueness implied by their construction. But I am convinced that the existence of the embedding proves (and it's not the only proof) that various AMPS-like arguments that the black hole interior can't exist in a consistent theory of quantum gravity – e.g. in string theory and its AdS/CFT – are just wrong.

The state dependence of the field operators \(\Phi(x,y,z,t)\) describing some fields in the black hole interior has gradually emerged as the epicenter of the controversy that prevents some physicists from confirming that Papadodimas and Raju have settled the broad AMPS-like questions.

Many previous blog posts – e.g. in August 2013 and August 2014 – unmasked my personal certainty that the concept of state dependence of the field operators is right. We must choose a realistic subspace of the Hilbert space – states that differ from a reference state \(\ket\psi\) at most by the action of some simple enough polynomials of the local operators – and within this "patch" of the Hilbert space, the field operators work exactly like quantum mechanics demands. However, the field operators don't have a broader range of validity – they can't be well-defined on the whole Hilbert space. After all, even the topology of the spacetime is variable which means that there can be no "universal" coordinates parameterizing the spacetime.

Here I want to promote a more obvious argument why the state dependence is inevitable. We know state dependence from statistical physics and the state dependence of the field operators is just a translation of these facts into the quantum gravitational language via the standard Bekenstein-Hawking-like dictionary.

Let's begin. In quantum mechanics, we have various observables. Everything that can be measured by a gadget corresponds to a linear operator on the Hilbert space, we often say. Positions, energies, angular momenta, parity, you know this stuff. Is really every measurement apparatus connected with a fixed linear operator on the Hilbert space? What about the thermometer? ;-)

I really want to ask about the thermometer and the entropy-meter – although the latter isn't found in most households. Let's not use the language of the housewives. Clearly, I want to discuss these matters at the level of mathematics. The question is whether the temperature \(T\) and the entropy \(S\), among related concepts we know in thermodynamics (entropy density, temperature gradient, heat capacity of an object, and so on) correspond to fixed linear operators on the Hilbert space.

Temperature may be position-dependent and it's complicated for that reason. But what about the total entropy \(S\) of the physical system in quantum mechanics? Shouldn't there exist a corresponding linear operator \(S\) acting on any Hilbert space in any quantum mechanical theory? It may be tempting but of course that you will agree that the answer is No.

Von Neumann offered a formula for the entropy of a density matrix \(\rho\)\[

S = -k {\rm Tr} (\rho \ln \rho ).

\] It is not a linear function of \(\rho\) – due to the logarithm and the two appearances of \(\rho\). It is not even an operator on the space of matrices – it maps a matrix to a scalar. And the very fact that it acts on matrices \(\rho\) and not vectors \(\ket\psi\) is annoying. Fine, can we define a linear operator acting on the Hilbert space?

Take a state \(\ket\psi\). What is its entropy? We would like to adopt a definition such as\[

S_{\ket \psi} = k\ln N_{\ket \phi, \,\ket\phi\approx \ket \psi}.

\] In words, I mean the Boltzmann's constant times the logarithm of the dimension of the Hilbert (sub)space of states \(\ket\phi\) that are macroscopically indistinguishable from \(\ket\psi\). Optimistically, we could find eigenstates of \(S\), compute the eigenvalues, and express the operator \(S\) with respect to this basis.

In practice, that won't work, of course. I said that we wanted to look for "eigenstates" of \(S\). That choice has already improved things a little bit. If \(\ket\psi\) were a general state, it would be extremely hard to count the number of "macroscopically indistinguishable" states from \(\ket\psi\) because \(\ket\psi\) is usually not macroscopically indistinguishable from itself! ;-) What I mean is that the typical states of large objects (and we are talking about large objects if entropy and other concepts makes any sense) are of the form\[

\ket \psi = 0.6 \ket{\rm dead} + 0.8 i \ket{\rm alive}

\] i.e. Schrödinger's cat states. They are superpositions of contributions that are macroscopically different from each other! So it makes no sense to talk about "macroscopically indistinguishable states from \(\ket\psi\)". What would that mean? Would we pick the dead cat only or the alive cat only? And if we tried to allow "cousins" of both terms, with the same coefficients, there would be many more terms that could be modified in various ways. And the ultimate problem would be that even in the purest cases you can think of, the "macroscopic indistinguishability" is not a sharp Yes/No question. States become indistinguishable gradually. And there's always room for conventions "how close they should be", and so on.

So there is a severe problem with the desire to define this entropy operator \(S\) in the most general large Hilbert space of a quantum mechanical theory.

However, there can be similar operators that play the role of this operator \(S\) in sufficiently well-controlled situations. Realize that your system is gas or a collection of qubits with a big degeneracy – you will be able to find pretty good operators that deserve to be called the entropy operator \(S\). I leave the examples to you as an exercise.

Even though most of us know the thermometer, the construction of a linear operator \(T\) is equally subtle or impossible. Roughly speaking, the temperature is the energy per unit entropy or one degree of freedom (divided by \(k/2\)). But the point is that the entropy or the "number of degrees of freedom" are hard to define – that's what the previous discussion was all about – so you can be sure that the temperature operator will face the same problem.

How is it possible that the thermometer works even though there is no canonical temperature operator on a general Hilbert space? Well, it has to work. It replaces the original system with the original system with a thermometer inserted, they interact in some way, and what we're interested in is the final height of the mercury which is given by a linear operator. Quantum mechanics is guaranteed to be able to produce probabilistic predictions for the height of the mercury. In any normal situation, you will also be able to show that the mercury is linked to the temperature in the usual sense.

Lots of questions arise at every step. I am sure you want to throw them at me en masse. How is it possible that the average height of the mercury has a linear operator and the temperature doesn't if they are the same thing? Well, they are not quite the same thing. The mercury–based thermometer, when exposed to the sunlight, will measure a higher temperature. That's why meteorologists prefer to tell us the temperature in the shadow. And if you use the thermometer to measure temperatures in tens of thousands degrees or more (e.g. inside the Sun), it will melt and the mercury will evaporate and its height will no longer be showing the correct temperature. These are two examples of a seemingly technical problem but these technical problems actually decide about the existence of the linear operator.

That will be particularly true in quantum gravity. In some sense, my argument will be equivalent to the claim that quantum gravity doesn't allow completely resilient and universal thermometers. ;-)

I have already mentioned it but the desired definition of the entropy operator has to depend on some "tolerance level" – how much different states may still be called "macroscopically indistinguishable". One thing to notice is that the threshold (or a more gradual treatment that replaces it) is a matter of a personal choice, a convention left to you. But there's another problem aside from this "subjectivity". The word "macroscopic" refers to large distances and we need some measure of large distances – something like a metric tensor. We want the occupation numbers for small values of \(k\) to matter; but we want to overlook the occupation numbers for some very high, generic values of \(k\). Different values of \(k\) are treated differently – and you need to have the metric tensor to distinguish the "relevant" long \(k\) modes from the "irrelevant" short ones.

You are invited to analyze all these problems and their possible resolutions and answer the question how general the resolutions are and what they depend upon.

But I really want to make statements about quantum gravity. Imagine that you have a quantum mechanical theory with black holes and you consider some states of several black holes in your spacetime. Some people believe that some approximate field operators on this Hilbert space may be defined in a state-independent way. If it is true for the invariant information carried by the metric tensor field, we should also be able to define the "state-independent" linear operator \(A\) of the total area of the event horizons of all these black holes, right?

But as Bekenstein and Hawking realized,\[

S = \frac{A}{4G}

\] holds in this case. The area is the same as the entropy in this situation with black holes. However, there's no nice operator for \(S\) as I discussed before – so there cannot be any equally nice, universal (acting on the whole Hilbert space) operator for \(A\), either. It means that there must be a difficulty to construct some invariant information about the metric tensor field. The acceleration (some Christoffel symbols) near the horizon are linked to the temperature which is also hard to define as an operator. The same difficulty probably applies to all other fields in the spacetime, too.

If we quantize gravity semiclassically, we seem to be able to identify the linear operators associated with the fields. And even a problem like the "proper area of some surface" that is described by some condition could perhaps have a well-defined operator. (Even this is actually untrue, but I don't want to go into that here because the problem we encounter outside the semiclassical treatment is more visible.) So why do we seem to have some operators for the metric tensor in the semiclassical approximation and not in general?

Well, because it's semiclassical. More precisely, it's because we chose a reference background geometry. This background geometry allows us to organize the "nearby" states in the Hilbert space and decide which of them are macroscopically similar to others. Different reference backgrounds imply different rules about "what state is macroscopically indistinguishable from another one"; think about states of entangled or non-entangled two black holes in the ER-EPR correspondence.

Black hole interior: it gets tough

All the problems with the definition of the "entropy operator" may be considered "nonzero but relatively minor" up to the moment when black hole horizons and especially black hole interiors enter the stage. Just to be sure, the operator surely doesn't exist in some canonical, conventional-independent way. But you could construct some approximate operator which could deserve the title "entropy operator" at the same level of morality as the usual (also non-canonical) definitions of the entropy in classical physics, for example.

But if black hole interiors appear, the hurdles become really brutal. Why?

Simply because all microstates of a black hole with a fixed mass, charges, and angular momenta must be considered macroscopically indistinguishable from the viewpoint of the outside observer! The black hole interior is invisible for the outside observer; which is why features of the interior can't make the microstates distinguishable. A stabilized black hole always looks the same – all the interior degrees of freedom are the irrelevant noise that the notion of entropy was invented to overlook. And that's a claim that the infalling observer simply cannot accept.

Because he can measure some local fields and similar observables in the interior, the infalling observer is able of distinguishing the black hole microstates macroscopically much more finely. You know, some microstates contain a beautiful woman whom the infalling observer loves and who will spend the last moments with him. Some microstates don't contain such a woman. The infalling observer may see whether there is a water company inside the black hole, whether he is the CEO, and if the infalling observer is a stellar physics instructor from a university in Cambridge, Massachusetts harassed by the bureaucratic establishment of the school, he can even measure whether the queefing is hers.

So how can someone argue that all the microstates are macroscopically indistinguishable? They are highly distinguishable from his point of view.

We must ask: How many similar points of view exist for a given black hole, a given large collection of black hole microstates? How many ways are there to split the Hilbert space to classes of macroscopically indistinguishable microstates?

The infalling observer can't make enough measurements to distinguish all the microstates, of course. But even if he could, we wouldn't expect him to do the "totally precise measurements". The very point of entropy is that we only talk about some fuzzy measurements, not the most complete ones. So the classes of macroscopically indistinguishable microstates will still be large – not far from the dimension equal to \(\exp(A/4G)\) – even from an infalling observer's viewpoint. He may distinguish "many classes" but the precision still preserves the huge size of these classes.

Is the splitting to the classes shared by all infalling observers?

Of course, if we talk about conventional humans, the answer is No. An infalling observer is human and can only measure the field operators in the vicinity of his body – and those he can easily see from his place. Lots of the information will be invisible from his place. He is not covering the whole surface of the black hole. He can't see everything.

You may object: In normal statistical physics, we were defining the "macroscopically indistinguishable" relation with the assumption that an observer can see all the macroscopic differences, so why did I just change the rules of the game and I separate the features that an observer can see from those that he can't?

I have to do it because of the black hole complementarity. In non-gravitational quantum physics, there was just one point of view – or all individual observers' points of view could have been extended to a shared, complete one. In quantum gravity, there are many points of view and they are complementary to each other. I can't talk about the indistinguishability according to both "all degrees of freedom outside" and "all degrees of freedom inside" because the latter are a subset of the former and the two ways to divide the Hilbert space into classes would be incompatible with each other.

As you can see, I am convinced that there will be many viewpoints. If a black hole is really long-lived, different observers may cross its event horizon at totally different times. And I think that it's obvious that these observers will have totally different ideas about how to split the Hilbert space of the black hole microstates to the classes of "macroscopically indistinguishable" elements.

Although I am not sure whether it's clear to Susskind and others, the statement that a black hole is a fast scrambler is just another way of saying that the arrangement or organization of the black hole microstates to classes of "macroscopically indistinguishable" elements at time \(t\) is totally different from the organization at a time \(t+\Delta t\) where \(\Delta t\) is a scrambling time.

It follows that all these observers swallowed by the black hole at different times will end up with totally different splits of the Hilbert space to classes of "macroscopically indistinguishable" elements. They will be recycling the same information in many different ways.

If a \(d=4\) black hole has mass \(M\) in Planck units, its entropy is of order \(M^2\) while the lifetime is of order \(M^3\). To be safe, pick an observer whose size is the geometric average of the Planck length and the initial black hole radius. In the Planck units, the observer's size is \(M^{0.5}\). It's clear that before the black hole evaporates, you will be able to throw \(M^{2.5}\) such observers into the black hole. Even though I made them rather large, it's still more than the black hole entropy in bits. Obviously, even if each such observer swallowed by the black hole only saw one bit, we know that the bits can't be independent – the Hilbert space can't be expected to contain the tensor product of all these qubits Hilbert space – because the number of these observers' qubits is just too large for the black hole entropy. (The used number of medium-sized observer was larger than the number of Hawking quanta which is of order \(M^2\); they are very slow, low-frequency particles.)

So of course that the individual infalling observers' qubits – and local fields in these observers' vicinity, if they are inside – are recycling the quantum information of the black hole many times in many different ways. These field operators can't be mutually commuting – not even when you pick the points of the observers' world lines that are spatially separated from each other. The spacetime in the black hole interior is just too large for that.

Papadodimas and Raju aren't able to tell you which of the embeddings of the interior field operators is the "right one" – the one that is relevant for predicting a particular infalling observers' observations. But that must have been expected because there are just many ways to do embed the operators. The rhythmically devoured observers above are just the simplest example of the multiplicity of the solutions.

I personally think that in principle, the observations of any infalling observer may be predicted probabilistically but otherwise uniquely if we know the exact initial microstate of the star that collapsed to the black hole. To do so, we have to organize the huge Hilbert space according to a background geometry that does include some parts of the black hole interior that the infalling observer will be exposed to. With a fixed background like that – which can never be picked uniquely, but it is selected if the problem is well-defined – string theory may be imagined to be a form of string field theory (that should only be used in a "diamond" or some restricted region to avoid conflicts with the black hole complementarity). Despite all the stringy physics and extended objects upon the background, the probabilistic predictions for field operators – which are qualitatively the same inside as they are outside, if we choose this description – may be done just like they are done outside or just like they are done in quantum field theory. Some representations of the stringy Hilbert space don't look like a "generalized QFT" in a spacetime – but I think that with a fixed reference background, the relevant portion of the Hilbert space may be parameterized in a QFT-like fashion.

But a state-independent definition of the interior field operators cannot exist because there's no state-independent way to split the Hilbert space to classes of "macroscopically indistinguishable" elements. For that reason, entropy-like and temperature-like quantities can't be written down as universal expressions. And because entropy and temperature is identified with areas and accelerations – functionals of the metric tensor field (and perhaps other fields), the no-go conclusion holds for the field operators (especially in the black hole interiors), too. We must always work with a "modest Hilbert space" around a particular state that describes the realistic excitations, excitations that are mild enough not to obscure which states are macroscopically indistinguishable from others.

And that's the memo.


  1. Very nice post ;-)

    I have a lot of naive questions...

    (Even in Semi-Classical GR different observers disagree about the vacuum)
    Does a theory of Quantum Gravity have a vacuum state?

    Does the picture we have in usual QMechs/QFT about a Hilbert/Fock space really pass on to Quantum Gravity if the vacuum concept is not well defined?

    For example in order to excite some state say to create a particle we need some energy/ where do we get it from?

    We can couple our system to another system that is in an excited state and we can destroy this other particle and then create one in our system, but if we want the total thing?

    One can argue that vacuum fluctuations can create pairs of particles but since all couples to gravity can one argue that all pairs are created from excited gravitons?

    Can we define (say for simplicity) a finite dim "Hilbert space" such that it does not have a lowest weight/vacuum state but is a circle- If we keep on exciting we arrive at the state we started- and then a "Fock space" with two types of particle populations (say blue + red) such that if we destroy one red we have to create two blues at the same state supplemented with a rule to jump from a state to the next one like removing one red from a state-put it in the next one and then we take the Union of all the sectors that contain from N reds-0 blues to 0 reds-2N blues for each state say such that the total max number of reds is fixed?

    Can we define approximate notions of vacuum in this case?
    (states with 0 blues)

    Does this construction implement any UV-IR connection?

    Thank you for the fun and interesting blog!

  2. It is interesting to know how difficult theoretical study of BH is.
    One immediate question (naïve): if there is no nice operator for A, then even
    the location of event horizon is uncertain (probabilistic) and depends on the
    observer. Is this correct?

  3. I should add that, the reason I am not sure, is that black holes we are talking about are macroscopic objects.

  4. I am not happy with a centralized controlled bank like the FED either, but it is now clear to me that Bitcoin is not the answer.

  5. This doesn't probably deserve an answer, but I was thinking about something your friend Nima said in one of his lectures, namely, that a good-enough physicist would be able to deduce most of modern physics with quantum mechanics (as formulated before Dirac I presume) and special relativity alone. My question is, could quantum mechanics (as formulated by Heisenberg and Shroedinger) have stood by itself as a consistent description of nature in the absence of special relativity?

  6. Dear Lubos,
    I like this article a lot. Maybe one question: The collapse of a star into a black hole seems to be a process, where classical physics seems to be good enough to describe the rough outcome and the correct final background geometry uniquely. So this "picking" of a background geometry must be related to the question whether one is an infalling observer or an outside observer. Could one maybe create a toy model how this "picking" comes about. Could the most fundamental obstacle perhaps be the special role of time e.g. in the state vector psi(t).

  7. The referred-to "state dependence" is yet another pain for the common sense portion of my percEPTiveness to have to put up with!
    Thus, to me, the self-discipline demanded by the fundamental QM-perspective implores or *is* a special sort of self-directed and 'self-applied' Mind Qontrol. %-}

  8. Everybody knows by now, it's right about time for a post on the following geometrical picture, EPR = ER = ESP (Yes, extrasensorial perception).


  10. I just read few chapters of 'State-Dependent Bulk-Boundary Maps and Black Hole Complementarity' and browsed through the rest.

    Lubos, I see that the authors express special thanks to you so I gather that, beside tending this blog, which seems like a full time job to me, you are very active in that area. Glad to hear about these topics straight from the horse's mouth.

    On to learn about the thermofield double. There is so much one needs to know about and understand at least in principle, to get a decent overall picture of what Kyriakos and Suvrat are talking about in this article.

  11. Dear Mikael, you are welcome.

    Well, my view on this question of yours is rather clear and I think that I wrote it, perhaps in different words, in this blog post, too.

    One may choose the "background" to describe a physical process almost arbitrarily - and get to any (or almost any?) desired state by the creation of a sufficient amount of stuff on top of the predetermined background. A curved geometry is a flat geometry with some gravitons and perhaps other things created on top of it, and vice versa, and infinitely many similar things hold.

    Classical physics or effective field theory describes what the background has to look like for the "amount of excitations and deviations" to remain minimal at all times.

    So the picking of internal/external arises because that's the part of the question, not a part of the answer. An external/infalling observer wants to answer questions about his vicinity, so he wants to pick a reference background for which the degree of excitation will be minimal. Even highly rough approximate laws are enough to see that at the beginning, some objects will survive outside the star, some will collapse to a black hole, so they may be used to basically estimate possible fates of people. And that's why we know what background is appropriate for both observers, what kinds of detailed questions may be asked.

    My point is that the fundamental physics will not contain a "completely rigorous" derivation of one branch or another because they describe the full physics just approximately.

  12. Yes, I think it is correct! Of course, there will be approximate operators and the spread of different ideas where the horizon should be will be limited. But depending on the observer and which states and ensembles of states he considers relevant, there will be somewhat different locations of the horizon.

    There is no measurement apparatus that would exactly beep when "we just crossed the horizon". It can't be measured this accurately, so that's we shouldn't expect any exact operator. In fact, in a general spacetime with a collapsing star, the location of the event horizon isn't known a priori even approximately: things happening far away may still influence whether any black hole will be created at all!

    The precision of the position of the horizon is only given by the same reconstructions and predictions of the spacetime using classical GR. And classical GR is just an approximation; and it's just a choice of a background which isn't canonically given for a microstate, as I emphasize in this blog post and related ones as the main point, more or less. The choice of a good background is just a practical choice - attempt to reduce the amount of "excitations above the background" that are needed to describe the states of interest.

  13. Thanks, Apeiros, for your interest!

    A theory of quantum gravity - e.g. string theory (well,probably, the only example) - has a vacuum state. It has infinitely many vacuum states. The "infamous" 10^500 of them are stabilized semirealistic vacua (vacuum states). It's the landscape. Such states are empty because of their being maximally symmetric - AdS or Minkowski (it works for de Sitter spaces only naively). That proves that there are no excitations on top of them. The Minkowski or AdS energy is minimized for these states. All other states in the Hilbert space have higher (eigen)values of energy - in the AdS, Minkowski case, which is why the ground state is the ground state.

    For de Sitter space - whose spatial section is effectively sphere, in some coordinates - one can't define the overall energy because there is no asymptotic region. So the "ground state" (state of lowest energy) is similarly ill-defined in the exact treatment. Semiclassically, one may find some vacua. But in the exact theory, it's more correct to say that there is a huge number, exp(A/4G) where A is the surface of the cosmic horizon, states that look pretty much like the vacuum state.

    The Fock space of QFT isn't an accurate description of the Hilbert space in quantum gravity. Such a space would be infinite-dimensional. But there is only a finite entropy one can squeeze to a region in QG. If there's too much matter, it gravitationally attracts each other, and we can't surpass the black hole "densities" etc., so at some moment, the higher levels in the Fock space break down and cease to exist in the conventional sense. Even before that, the stucture is curved because the geometry backreacts. So the Fock space is only a good approximation for a QG Hilbert space is the degree of excitations (density of matter, roughly speaking) relatively to the chosen background (geometry plus field) is small enough.

  14. IMHO black holes could have stranger habits than moet people think if fermions are little string propellers with double spin axis..
    According to Quantum FFF Theory, there is no fermion information loss, only photonic information loss in a black hole.
    Quantum fluctuations around the BH horizon produce pairs of leptons. e-,e+ and even compound quarks (d,u, etc, see Fermion 3D string propeller theory: Fermion repelling by Lorentz polarized spin flip). ( see also Vixra "mass in motion")
    For tripple BH horizons see image nrs 7, 10 and 11. Event horizon=7, Inner photon ring = 10, Outer photon ring = 11.
    Conclusion The Big Bang did not produce instantly all the Fermions in the universes. Even now lots of Fermions are produced- in the form of negative charged plasma- by all BHs (see ball lightning) even the largest primordial Big bang splinters located outside large galaxies. see also:

  15. Thanks for the thorough answer, it is very helpful!

    Some comments i want to make since yesterday I was half asleep ;-)

    I was trying to find a simple QMech model that shares some of the properties of QGR...

    1) For the discrete time

    I had in mind the old \(c=1\) matrix models where Gross/Klebanov had shown that the dimension where we define the matrix model (not Liouville direction) can be discrete as well as continuous and there is no difference as long as the discrete points are close enough \(\epsilon \le \epsilon_c \).

    I do not know what holds in (M)atrix theory and subtleties if we wick rotate this discrete dimension...

    2) I totally like your digression regarding vacua in string theory, but is this picture really quantum mechanical or some semiclassical view?

    3) About the Fock space, yes QGR should not have a Fock space as in the normal QFT definition, and this is why I put "Fock space" in the model I described, since this construction has a finite number of states (say M states on the circle and each one can be populated by up to N reds or up to 2N blues - in any case the total number of states in the theory is finite- it seems to me that also in QGR we cannot create an infinite amount of particles)

    4) Since we have a circle I think it is not legitimate to speak of high and low energies, so in this sense i was speaking of UV/IR connection

    5) I was wondering whether we can still have an approximate notion of ground state and UV/IR, since this model interacts and we will find some of the M states that are mostly populated by either only blue or reds or not populated at all and if vacuum is for us the most empty state, I think we have an approximate notion of vacuum as well as excited states, that can change as the system evolves and interacts....

    6) I made a clear mistake when I wrote that a particle can emerge from the vacuum.... sorry for that....

    7) Should not a theory of QGR have 0 total energy?

    8) In QGR as you said if we pick a vacuum (in my model an approximate vacuum) and then we start forming particle excitations we form an approximate Fock space that breaks down when we discover that particles have to backreact on the geometry or equivalently to create a lot of gravitons (is this equivalent to treating say reds as gravitons and blues as particles and then disregard the fact that reds couple to blues up to the point that we put more blues in our state that we can - we have an upper limit- and then we discover that we need to treat reds also....)

    In any case some points are not that clear to me so any critique is welcome ;-)

  16. "On linear operators to determine who is queefing"
    A contribution to the XXX archive by LM.

  17. Dear Luke, even more so than string theory, quantum mechanics is a framework, a set of principles. It doesn't determine the law describing Nature completely.

    To do so, one has to choose the Hamiltonian - or some other relevant operators on a Hilbert space. The right realistic Hamiltonians are just "some choices".

    The non-relativistic quantum mechanics for a few particles is an approximation that holds at low speeds etc. At higher speed, one needs to take relativity into account. In the context of quantum mechanical theories, it means that one has to consider theories that are both quantum and relativistic - something like quantum field theory or string theory. Otherwise your description of Nature will be wrong.

  18. Sorry, it's just too much stuff that exploded exponentially. I can't answer such long series of questions.

    Just to be sure, energy measured asymptotically in GR, like ADM energy, search for that, is nonzero in general.

    The stringy landscape is an exact statement about some Hilbert spaces. They do exist in string theory, no doubt about that. One may mostly separate the Hilbert spaces built on different asumptotic conditions at infinity to different superselection sectors, isolated from others, so the 10^500 choices label the number of superselection sectors.

    The circle may have a preferred origin etc.

  19. Dear Lubos,
    whether the life of the astronaut ends in the forming black hole or he can escape and live a peaceful life with his family looks like a perfectly well defined physical question to me. And with a perfect knowledge of the quantum state of the star and the astronaut quantum theory should be able to make a probablistic prediction for this.

    What you seem to say is that we need the less fundamental theory (GR) to make predictions in the more fundamental one (quantum gravity). I can accept that this is a limitation of the current pertubative methods or perhaps even of the current formalism of quantum theory. But a limitation of fundamental physics?

  20. Dear Mikael, I don't think that the picture I presented implies any incompleteness of the theory of quantum gravity.

    And your suggestions that this implies some incompleteness look similarly fallacious as claims that quantum mechanics is incomplete. In fact, the QG case is a special example of the QM case.

    One may predict probability distributions of a particle position; or its momentum. But one can't calculate the probaility distribution for both at the same moment because the two things can't be measured simultaneously.

    So even if we have a complete quantum mechanical theories, one must first exactly specify what the question is. The definition of a background to build the Hilbert space as a Fock space or something close has to be a part of any fully well-defined question.

  21. Marcel van VelzenDec 14, 2014, 12:08:00 PM

    "One of the main benefits of Bitcoin is that it is not subject to
    centralized control" that's also (more or less) true for gold so you
    don't really need bitcoin for that. It is more the combination of this +
    best known money for the Internet.

    "that will change as the availability of mining hardware increases and
    its price comes down" At least in the Netherlands it is very easy to buy
    state of the art mining hardware, so that's not the problem. The real
    problem is that you need to get the hardware and the energy at the
    lowest possible price which is now only possible for professional
    miners. The general rule is "don't mine bitcoins but buy bitcoins". I am
    mining bitcoins only because it's the best way to learn how mining
    really works and because it is fun to have mined say 1 bitcoin, but it
    costs me about a few hundred euro more per bitcoin than buying.

  22. Well, my question is, will a particular astronaut near the collapsing star fall into the black hole or not?

  23. You should have titled it:
    "Undergraduate Review of Last Centuries Keynesian Thinking"

    These arguments where already outdated long ago by Milton Friedman, Hayek and Mises. The modern discussion has long moved passed the points discussed above.

    At least you got time value of money right and the point about the intrinsic value of gold.

    Yes, gold is dead and there still might be some callers in the dark, who want a gold backed currency back, just as there may still be some around who want the king back.

    You don't seem to understand what fail allocations of money and resources are caused by central banks setting the money supply and the interest rates. How many bubbles do you need until you understand? What happened during the DotCom bubble? What caused it? What caused the housing bubble? What is inflating housing and stock prices today?
    Yes, we are right in the next bubble and you don't even seem to understand what the underlying problem is.

    So 10% inflation is okay? It's only an isomorphism? True, any amount of money is okay, but what about the 10% printed and spend by the government? This is a distortion to the markets and suboptimal use of resources.

    This is like some anti nuclear greenpeacer pushing the standard popular arguments. This is like some climate change fanatic repeating the usual junk.

    Perhaps you should stick to the things you understand of read up on some economics, because otherwise you will be doing the Noam Chomsky.

  24. During the war, they fed and housed revolutionary troops, and one of my male descendants was a deacon of the church which was attended by George Washington, which makes me eligible to join the Daughters of the Revolution. Several of aunts are members of this organization. They have encouraged me to a join as well, perhaps because I'm a bit of history buff and enjoy collecting antiques. But I'm not interested in joining such an organization for the same reason I was never interested in joining a sorority or becoming a cheerleader. Even though I'm very nostalgic and love connecting to the past, anything that smacks of tribalism and snobbery turns me off. And I don't want to join because I'm not patriotic. Quite the contrary. I'd simply rather spend my time opposing imperial wars and Wall Street corruption. i

    Nevertheless, not long after the Revolutionary War, my ancestors moved west and settled in what is now the northwest corner of Missouri. They were true pioneers in every sense of the word. It was a very tough life for them but they endured to become fairly prosperous farmers in the region, enough to own a few slaves in fact. Though being so far from the North vs South epicenter, their involvement in the Civil War remained minimal. That is perhaps a good thing, my black and northern friends can't ever accusing me of being sympathetic to the Old South.;~)

  25. While there is a lot of family history, my parents raised us as a bunch of heathens, 6 kids on a farm and being 5th, I grew up fairly feral. They tried sending me to church, but it didn't take. The interesting part is they both grew up in fairly respectable conditions; My mother was a Warfield and grew up in the Maryland equivalent of Tara, out in Howard County, as her grandfather had been governor and her father was publisher of the Baltimore version of the Wall Street Journal, called The Daily Record. My father was a classmate of George H.W. Bush, but there were more crazy relatives than money on that side of the family. I ran away several times as a teenager and managed to spend a night in jail, in Lebanon Missouri. My wife was a Catholic school teacher, of all things, horses were the connection and my daughter is much more focused on climbing to the top of the pile in life. Not because she doesn't understand the options and consequences, but because she is a very driven person. She has been told by a number of people since she was a kid that she would make a good lawyer and while she resisted taking that direction, seems to be going there.
    One of my early insights in life is that what defines us, also limits us and so if you want to be able to see the big picture, you have to be able to step back from all that would naturally draw in your attention. Be it everything from religion, to money, to beauty. Our minds are like magnets and our perception of reality is like a bubble and the consequence is that we naturally get lots of stuff stuck to the surface of our bubble of perception and it becomes us and we flow naturally around in it, but if we want to keep growing, we can't get too obsessed, focused and defined by any one part of it. Religious texts are a good example of how society tries to get everyone thinking the same thoughts and marching in the same direction. Politics and nationalism come in a close second, especially when they are framed in terms of what we are for and against. That's why in conflicts, often those most reviled are not ones on the opposite side of the issue, but those who don't take sides, because they are the ones really questioning the whole framing device. Often there are two legitimate sides to an issue, because reality is that dichotomy of energy pushing out and order and form pushing in. So in situations where the social order can not effectively channel the energies of the people, it does get oppressive and so the people do push on the weak spots of the structure, which only inflames its control and policing functions. Too much order and you have North Korea. Too little order and you have Somalia. So it has to fluctuate between the two.
    Hopefully we are not pushing Lubos' patience.

  26. Dear Mikael, this question may be addressed from various perspectives. For the exterior observer, it may be asked whether the astronaut exists alive and compact at some time much longer than the black hole radius after the black hole is created. From this perspective, it's settled in a way that doesn't assume the existence of the black hole interior at all.

    In the astronaut frame, one may discuss more detailed questions about his observations even after crossing the event horizon, it he happens to do so.

  27. It seems that once the zero-law (and so the whole idea of thermal equilibrium) is at risk when living out of the semi-classical realms, then there is not much sense on continuing with those handswaving arguments on state-dependent operators for temperature or entropy. Without a globally well-defined temperature of equilibrium, no further laws of thermodynamics can be postulated, and so the thermostatistics underlying black-hole thermodynamics seems to be just an "evaporating" concept.

  28. As always, thanks.

  29. Thanks for that, too.