Thursday, February 24, 2005

The entropic principle

The anthropic principle has a new competitor.
It's called the "entropic principle" even though Hirosi Ooguri, Cumrun Vafa, and Erik Verlinde finally did not use this entertaining term for the title. Let me first state a popular version of the principle for those who don't expect to follow the details of this article:
  • The probability that the cosmological evolution will end up as a Universe with a particular shape of the hidden dimensions (and particular values of the fluxes) is determined by the (exponentiated) entropy of a corresponding black hole whose geometry flows via the attractor mechanism to the given shape of the Universe near the horizon. Note that this contrasts sharply with the "anthropic principle" - which itself is not a principle, rather a lack of principles. In the anthropic principle, the corresponding probabilistic weight is determined by the ability of the Universe to support intelligent life.
Topological string theory allows one to calculate the partition function which is a function of the complex structure moduli of a Calabi-Yau three-fold. Except that it's not quite a function; because of things such as the holomorphic anomaly, it is a "wave function". What do I mean? You may think that it is a function that only depends on the holomorphic moduli. Naively, you would expect that the function is holomorphic - it depends on X but not Xbar.

However, an infinitesimal change of the reference point in the moduli space induces the so-called holomorphic anomaly which a slight, exactly understood dependence on Xbar that can be locally visualized as an infinitesimal Fourier transform. By an infinitesimal Fourier transform, I mean the conversion of a wave function "psi(x)" to "psi'(x')" where "x' = x+epsilon.p" - you see that we are mixing "x,p," the coordinates on the phase space. Therefore it is more appropriate to talk about the partition sum as a vector in a Hilbert space rather than a well-defined function.




Also, the partition sum is a complex number whose squared absolute value has been shown to have many physical interpretations in various recent papers that had Cumrun Vafa among its authors; it has been related to black hole entropy as well as the partition sum of two-dimensional gauge theory. That's another reason to imagine that the partition sum is a "wave function".

In the new paper, this concept is taken very seriously. "The wave function" is interpreted as nothing else than the Hawking-Hartle wave function of the Universe. You know, the Hawking-Hartle wave function is something like a wave functional of quantum gravity that solves the Wheeler-deWitt equation (a sophisticated definition of the quantum equation
  • H.psi = 0
that is appropriate in general relativity but whose exact meaning requires a working quantum theory gravity, i.e. it requires string theory). The Hawking-Hartle wave function is a functional of the fields of quantum gravity on S^3, if you allow me to deal with the "most realistic" example, and this functional may be calculated as the path integral of quantum gravity defined in the ball B^4 inside this S^3, with the right boundary conditions at the sphere S^3. The Hawking-Hartle state is then the functional of these boundary conditions.

In the case of string theory, we may be afraid that the Hawking-Hartle state is a functional of a much broader collection of fields, including excited strings. Ooguri, Vafa, and Verlinde only consider a restricted set of degrees of freedom - a minisuperspace. They only consider the Hartle-Hawking wavefunction to be the function of a few parameters that describe the shape of the Universe - namely the complex structure moduli of the Calabi-Yau manifold. This truncation should be enough to calculate all holomorphic, SUSY protected quantities. The Universes they study have the form of AdS2 x S2 x CY3 and the corresponding Hartle-Hawking state is defined on S1 x S2. This may look like an oversimplification, but the advantage is that the simplified Hartle-Hawking state may be calculated including all the higher-order corrections - it's calculated as the partition sum of the topological string theory on the same Calabi-Yau three-fold.

The results may be viewed as something that gives you a "natural" probabilistic measure for the Universe to have different particular values of the complex structure moduli. Of course, their context only deals with AdS2 - a space that is geometrically equivalent to dS2 - and they're using both directions of this space in the role of time in various viewpoints that they relate to each other. In other words, they apply a double-Wick-rotation on their AdS2, and they're slicing it in two different ways, in analogy with the open-closed duality seen for the cylindrical worldsheets in string theory.

You may argue that the supersymmetric AdS2 backgrounds are not a good description of our Universe, and you may be right. But there could exist some generalization that will turn out to be relevant for the vacuum selection problem sometime in the future.

At any rate, I like their brave attempt to replace the naive "vacuum counting" strategy - a prejudice that each single "vacuum" of string theory is supposed to be equally likely - by something more realistic and potentially justifiable by quantum cosmology. This attempt is one of the first ones that uses the Hartle-Hawking wave function; I was informed by a reader-insider about an earlier paper by Firouzjahi, Sarangi, and Tye from the last summer that tries to phenomenologically apply the Hartle-Hawking state on the KKLMMT compactifications.

The Hartle-Hawking states are very natural and important tools that may eventually shed light on the vacuum selection problem and the evolution of the Universe a Planck time after the beginning ;-), and therefore I think it is a good idea to try to give the Hartle-Hawking wave function a well-defined meaning in string theory. Ooguri, Vafa, and Verlinde have certainly made many new steps towards this kind of goal.

The measure as they formulate it today kind of gives the different Universes a weight proportional to the entropy of the corresponding black hole. Although one should not be comparing different topologies of Calabi-Yau manifolds, you can nevertheless do it, and your conclusion will be that the Calabi-Yau manifolds with large fluxes - corresponding to large black holes - will be dominating the ensemble even more dramatically than in the "democratic counting".

I don't like this result too much and I hope that it will eventually go away, but this inconvenient preliminary result can't diminish my sympathy for the thesis that the attempts to define and calculate the Hartle-Hawking state are more scientific than the ad hoc assumptions that "every vacuum has the same weight".

42 comments:

  1. Hi Lubos,

    I have what will probably sound like a very basic question to you but I am not a student of physics.

    My undestanding is that Time is not an absolute measurement. If two particles are travelling at high speeds they would be also moving at somewhat different rates of time.

    My question is does the difference in time between the two moving particles remain constant or is it " elastic"? That is, does the variance in time change randomly/unpredictably as the particles move ? Or does it change in a predictable fashion depending on other factors ?

    I may not even be expressing this question correctly so anything you might care to say would probably be helpful

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  3. Hi Mark,

    in classical physics - this is the set of physical laws that were believed before the 1920s, the laws that don't include quantum mechanics - everything is completely determined.

    In physics, we must define meaningful things in such a way that they can be measured. So one defines "time" as the quantity measured by clocks. One can discuss what kind of clocks are we talking about, but it won't really matter.

    In physics according to Newton, there existed universal time that was valid everywhere in the Universe.

    According to special relativity, discovered by Einstein in 1905, the time depends on the reference frame, and moving clocks seem to go slowed by the factor sqrt(1-v^2/c^2) where "v" is the speed. This is a completely determined, deterministic effect of the speed on the "rate of time".

    In quantum mechanics that people have put together in the middle 1920s, phenomena are statistical and kind of random. One can only predict the probabilities of various outcomes of your experiment. This may be what you call "elasticity"?

    Even according to quantum mechanics, it is possible to construct arbitrarily accurate clocks that will measure the amount of elapsed time with arbitrarily good accuracy you want. You must produce the clocks using a right technology, and they must be sufficiently "large" so that certain probabilistic, fluctuating effects of quantum mechanics are suppressed, and so on.

    Generally, in physics, we always want things to be as predictable and exactly determined as possible. In classical physics, it was the case and everything was totally exact, inelastic, and determined. In quantum mechanics, the detailed results are uncertain, but the probabilities are again completely certain and exactly predictable - and verifiable by repeating the same experiments all over many times. And even in quantum mechanics, one can construct very precise clocks etc.

    But I've probably misunderstood your question completely.

    best
    Lubos

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  4. Lubos, I'm shocked! I'm surprised you take the Wheeler-deWitt equation seriously but not loop quantum gravity!

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  6. Dear Anonymous,

    I agree that loop quantum gravity may also be viewed as an attempt to make sense out of the WDW equation. Unfortunately it is not a consistent or successful attempt - simply because the Hamiltonian constraint can't be defined. (In fact, even the physical Hilbert space is not well-defined.) See an older article "A very meaningful paper on loop quantum gravity".

    All the best
    Lubos

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  7. But even the Wheeler-deWitt equation needs to be regularized!

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  8. That's right. This is why you need string theory to transmute the WDW equation from a heuristic speculation to a meaningful law of physics.

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  9. It's a huge jump to go from "the Wheeler-deWitt equation needs to be regularized" to "we need string theory to UV complete the theory". Ever heard of UV fixed points? Besides, there are many ways to UV complete a nonrenormalizable theory and it doesn't have to be string theory.

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  10. Hi Lubos,

    Seems like an interesting paper. I am wondering in what sense this is the Hartle-Hawking wavefunction rather than any other wavefunction.

    Speaking of which, did you ever understand what is the criteria for choosing any wavefunction (or initial condition) in quantum cosmology? as far as I can tell the only criteria is that the function is simple or natural in your favorite set of variables, not a very physical criterion.

    But then again, what would be such a physical criterion? usually choosing a state is arbitrary.

    You see, I am confused...

    best,

    Moshe

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  11. Lubos said:
    "Even according to quantum mechanics, it is possible to construct ARBITRARILY accurate clocks that will measure the amount of elapsed time with ARBITRARILY good accuracy you want. You must produce the clocks using a right technology, and they must be sufficiently "large" so that certain probabilistic, fluctuating effects of quantum mechanics are suppressed, and so on."

    See here is how Lubos making low level mistakes again. There is no absolute space or time. The time is ALWAYS associated with events that happen on time. You can not talk about time measurements unless you are talking about time associated with certain events you are trying to measure.

    Therefore, the theoretical possible accuracy of time measurement is ALWAYS dependent on what events you are talking about.

    How accurate your time measurement can be really depend on how well defined the events are in terms of time. For example, the exact time when Lubos will wake up this morning will not be well defined and so no matter what technology you use you can not determine that time to better than a fraction of second accuracy. And possible up to 10 minutes inaccurate if Lubos is a lazy sleeper.

    In quantum mechanics limit, the theoretical time accuracy really depend on the scale of energy involved in events. It's the Heisenberg uncertainty principle. For example, the exact time a photon of frequency 1x10^14Hz really can not be determined any better than 1x10^-14 seconds accuracy. If the photon emitted has higher energy, you may be able to determine the time more accurately, but still there is a limit.

    Further there is an absolute limit of the best time accuracy you could achieve, based on all accepted science today. That is the Planck Time, which corresponds to a mass/energy of one planck mass. You can involve a higher energy scale than Planck mass but it does not help you, because then you have a mini-blackhole and you can not determine the position better than the radius of the mini-blackhole, which is one planck length. Again space and time are really the same thing in relativity, so any inaccuracy in position measurement also translates into inaccuracy of time measurements. Worse, near the event horizen of the mini blackhole, time actually freezes and so it's not even meaningful any more.

    And finally, according to special relativity, no matter how accurate a time measurement you think you have achieved, I can always watch your measurements from a spaceship that flys at almost the light speed, and I see you are doing your measurements in a terribly slow pace and your measurements are not precise at all in terms of time accuracy. The more closer to light speed I am, the more inaccurate I find your measurement to be.

    So it's a complete nonsense claiming that theoretically you can achieve any arbitrary precision of time measurement. The total quantum information of the universe is a fixed amount and that puts an ultimate limit of accuracy of any physical measurements.

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  12. Anonymous: "It's a huge jump..."

    No, it's not a huge jump. As of today, there exists no evidence that gravity has a UV fixed point. It's one of the speculations that seem increasingly unlikely as the decades go and no one finds a realization.

    The only known consistent quantum UV completions of gravity in d>=4 are found in different backgrounds of string theory, and it is pretty likely that they're the only ones that exist.

    All the best
    Lubos

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  13. Hi Moshe!

    Thanks for your comment. I think we agree that the wavefunction they propose is definitely special. If nothing else, it's a unique state that appears in a paper by Ooguri Vafa Verlinde. ;-)

    But as far as I undestand, it's not just special in this sense. It is also the HH state computed in the minisuperspace approximation. The HH space is obtained from a path integral if you will the interior of the S^1 and compute the path integral, and I think that if you read the paper and look at the pictures, you will see that this is what they did.

    Do you have the opposite feeling?

    All the best
    Lubos

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  14. Hi Lubos

    Thank you very much for taking the time to explain time dilation effects. The word I was looking for to describe the effect I was asking about was " probalistic" and not " elastic" ( shades of Econ !).

    Now that I'm older I regret never having taken in a formal study of physics. There's a kind of thinking that is required
    (or learned) for fields dealing with analyzing complex dynamic systems -Physics or Economics being the prime examples - that other areas of study do not stimulate to nearly the same degree.

    Much appreciated !

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  15. Hi Lubos,

    First apologies, it is obvious from the paper why this state is analogous to the HH state.

    My confusion is much more general, there is more than one state one can write, even more than one simple state (e.g. the prescription by Linde and Vilenkin). One can even imagine writing an elegant expression for a state that will give a uniform distribution of the flux vacua. Why do you think then that the HH state is more "scientific", is there any physics reason to prefer it? is there even any physics way of defining it?

    best,


    Moshe

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  16. Dear Moshe,

    I see, so you're questioning not only the new paper, but also the meaning of the HH state in general.

    As far as I can say - sure, HH is extremely preferred on physical grounds. HH state is the state of the Universe that can be "created from nothing".

    The Universe was created from the Big Bang, and people ask What was before? We usually say it's a bad question, and the Universe can be "created from nothing" - it simply starts. But it's only the HH state that can really start in this way; other states require "singular insertions" at the beginning.

    Imagine that you study the worldsheet and you draw a circle in it and view it as space in radial quantization. Then the state corresponding to the identity operator - smooth history inside the circle - is the analog of the HH state. Of course that it's physically preferred.

    Inversely speaking, if you think that it's not preferred - which probably implies that you think that the HH paper is meaningless - do you have an explanation why they wrote the paper and why others discuss on this in their papers?

    Best
    Lubos

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  17. Lubos,

    The problem is, indeed, that many smart people like the HH state (including of course Hartle and Hawking), but many smart people do not (including Linde and Vilenkin), so sociological decisions will not do.

    The statement of being created from nothing is Euclidean and therefore seems like a mathematical criterion to me. Do you know any Lorentzian physical way of stating the special properties of the HH state,e.g. it is the
    most symmetric initial state etc.? with such statement one may discuss if the properties of this state distinguish it from others.
    (and I plead again ignorance, I do not know enough to make
    a judgement, but with your help I may).


    best,

    Moshe

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  18. Hi Moshe,

    we just briefly discussed your points with Cumrun.

    You know that I agree that the sociological criteria don't matter much, especially if they're undecisive.

    But would you agree with our statement that the HH state is the only known canonical prescription that can say anything about the initial conditions of the Universe?

    I completely misunderstand why you object to using the Euclidean path integral. It's a legitimate mathematical tool that gives the correct answers to hundreds of other questions - it's more well-defined mathematically and easier to use.

    The Lorentzian spacetime interpretation of the state is simply that the Universe starts at a particular state and then it evolves as usual.

    But this Lorentzian picture can't in principle say anything about the initial state because the Lorentzian metric itself breaks down at the very moment of the Big Bang anyway.

    The HH state is the most natural "ground state" of the Universe. You know, we don't know whether it's indeed relevant for the beginning of the Universe. But it is by far the most convincing mathematically justified canonical choice of the initial conditions of the Universe, and therefore it is natural to try to make sense out of it in a full theory of quantum gravity.

    All other choices that people have proposed so far are ad hoc religious guesses. The HH state is the only thing that is actually based on the real action/dynamics that govern a given theory.

    If you think that you have a better calculation (better than HH and its realizations) of the initial conditions in a given quantum theory of gravity, I will be happy to hear about it. Otherwise I think that the discussion may become vacuous. The HH state seems as the best canonical candidate for the initial conditions, and it's not guaranteed whether it's the right one. There's nothing more to say about it.

    Best
    Lubos

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  19. Lubos,

    I am honoured...


    I agree that the HH state is the canonical
    choice within the framework of Euclidean quantum gravity, and it is very useful to generalize it to string theory. It may also resolve known problems in making sense out of
    the HH prescription (e.g. the wrong sign for the scale factor), but I am out of my depth there.

    Of course any choice of wavefucntion is just that, a choice, but I was wondering if there was a precise way of characterizing that choice which does not rely on a particular formalism. Since the prescription is so simple, and it does use the dynamics of the theory, it seems natural that one can phrase that choice simply,and indeed that this was already done by someone.

    I agree that we reached the point where the discussion needs some alcohol boost,maybe we will continue this in person sometime.

    best,


    Moshe

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  20. Dear Moshe,

    we seem to differ in our opinion about the very basic question which kind of argument is convincing and which is not.

    I disagree with the statement that "any choice of the wave function is just another choice". It is a legitimate opinion to think that the question about the initial conditions of our Universe is comparably important as the question about the physical laws that govern the subsequent evolution. It is a scientific question that has one correct answer and many incorrect answers that are not interesting.

    If this question can be answered without adding anything new to the theoretical structure - and by using the original action (or equivalent description of dynamics) only - that's definitely a progress.

    Would you also say "any choice of the physical laws is just another choice"? I hope that you would agree that this would be an unscientific position. We definitely want to know which physical laws are the right ones and which are not, and saying "one choice is as good as any other choice" would simply be a sign of ignorance.

    You also call for "characterizing that choice which does not rely on a
    particular formalism". I don't sympathize with this either. A convincing answer to a question within a given quantum theory is exactly an answer that DOES rely on the formalism as opposed to other things. If someone proposes another answer that does not rely on the formalism and requires completely external choices and principles, then it's definitely less scientific in my opinion.

    This includes initial conditions that would agree with the Bible; initial conditions that would realize some particular description of "equality" between "some things" that can't be justified dynamically.

    The quantity called time is not a terribly good concept in the very early stages of the Universe because spacetime is as fuzzy as it is big at that moment. This is why the Hamiltonian quantization of the theory on sharp temporal slices is probably a bad formalism to apply at the early Universe. The path integral is more promising for the questions of early quantum cosmology, and the Euclidean path integral is the mathematically more well-defined way to obtain the relevant numbers and functions.

    All the best
    Lubos

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  21. Lubos,

    Thanks a lot,this is very useful, I am starting to see your point of view.

    Allow me to re-phrase to see if i understand - the idea is that the choice of wavefunction does not really exist, or at least it is not completely arbitrary. Within some non-perturbative completion of Euclidean quantum gravity most ways to start the universe would not make sense. The formalism then would predict the wavefunction of the
    universe the same way it predicts it subsequent evolution.

    Did I get your viewpoint correctly?

    best,

    Moshe

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  23. Hi Lubos,
    Have you seen this paper:
    http://arxiv.org/abs/hep-th/0410213
    The authors there seem to imply that the smallness of the cosmological constant can be explained without evoking the anthropic principle. They also use the WDW equation in their analysis. I think that the main problem with their setup is that the value of the cosmological constant is (not surprisingly) equal to the SUSY breaking scale.

    Kostya

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  24. hmmmm... I'm thinking that the probability that the cosmological evolution will end up in a universe like ours is determined by an "entropic" anthropic principle, which constrains the entropy of a corresponding black hole, whose geometry flows by way of the attractor mechanism, to the shape of the Universe near the horizon.

    ... but I have an inherent anthropic bias... ;)

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  25. Dear Moshe, thanks, it's very likely that you're interpreting the viewpoint correctly.

    Thanks to Kostya for his link.

    And thanks for keeping anthropic bias on the island. ;-)

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  26. You know the standard objections against Euclidean quantum gravity. It's not possible to Wick rotate a generic Euclidean manifold into a Lorentzian manifold and vice versa because most metrics lack a timelike Killing vector field.

    That's why Lorentzian quantum gravity is the way to go.

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  27. I know the "standard" objections", but the HH state is definitely an example where the Euclidean gravity is more fundamental.

    This reminds me of the paper by Bogdanov brothers which is reasonable in this sense. They argued that near the big bang, the theory becomes topological and even the signature is uncertain etc. Although the details how they connect these claims with mathematics are not quite serious, I agree with their general claims. Near the big bang, the usual notions of geometry, including the signature, must be modified. The Lorentz symmetry (and even signature) becomes useless.

    We know many examples like where the Euclidean path integral is simply superior. The calculation of stringy scattering amplitudes is usually done on a Euclidean worldsheet. It's much more well-defined and gives the correct results.

    I would guess that you come from the loop quantum gravity environment. You know, there the situation is very different. Loop quantum gravity is inconsistent in any signature, and moreover the different signatures can't be used to get the same results. They're not continuations of one another, not even in flat space (which is not present) - an example of an inconsistency.

    But in a consistent theory of quantum gravity, the situation is very different. All signatures are mathematically consistent, and they are equivalent with a proper choice of analytical continuation. The Euclidean approach is almost always mathematically superior.

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  28. Hi again,

    I am learning here, so I hope you will not be offended if I make a couple more comments, and ask for your opinion.

    The Euclidean approach is very useful for some things, but it is not without it's problems. Let me list two I am aware of.

    So, first a mathematical issue. The Euclidean path integral is not really that well-behaved. As you know the Einstein action is unbounded from below even when making Euclidean rotation. The scale factor has wrong sign kinetic term. The path integral can be defined perturbatively by appropriate contour choice in field space, but as far as I know (and I don't know much), the path integral was never given a non-perturbative definition. Of course this is one reason it is exciting to define it in a mathematically well-defined theory of quantum gravity, such as topological string theory.

    Secondly, a physics issue. When considering black holes, one can also use Euclidean rotation, the horizon is mapped to a point and regularity at that point is the statement of the Hartle-Hawking boundary conditions at the horizon. The physics is well-understood, this is the case where the hole is in equilibrium with surrounding radiation. In that case however it is evident there are other possible states. For example there is the Unruh state where there is no radiation and the hole shrinks. In the Euclidean theory the origin will now indeed become singular, but the reason is clear- the situation is time-dependent. I think in that example it is difficult to argue there is something unphysical
    with the Unruh state, the Euclidean formulation simply is not adequate for that particular situation.


    Finally, in string perturbation theory the worldsheet time is a Schwinger parameter, so the signature of the worldsheet is a separate issue. Interesting, but separate.

    So, there are very smart people (such as my colleague Bill Unruh, who has written about the subject) that are skeptical about the Euclidean formulation being more than a mathematical trick, useful is some circumstances. Maybe the situation of quantum cosmology is one such case.

    best,

    Moshe

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  29. Lubos, you know full well that Wick rotation works on a 2D worldsheet with a Weyl invariant metric and it's fully rigorous. The same thing goes for any conformal field theory.

    But a Euclidean/Lorentzian manifold in four dimensions is another thing altogether. Given any functional over Euclidean metrics (which typically diverges as the signature changes), there are problems with analytically continuing the functional to Lorentzian metrics.

    And you're wrong about the Wheeler-deWitt equation describing sharp temporal slices. Rather, it describes a timeless universe, or if you wish, it describes all times simultaneously.

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  30. If it can be shown one of the loop quantum gravity theories has a second order phase transition describing a weave state, that would be an existence proof of a UV fixed point for general relativity. Of course, the existence of a second order phase transition is still an open question but it's likely one would exist in analogy with other similar models.

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  31. Moshe, thanks for your new text.

    I don't understand the origin of your comments about giving an exact meaning to the quantum gravity path integral. The path integral of quantum gravity only makes sense in a consistent theory, i.e. string theory. Do you agree that there are 2-loop non-renormalizable divergences in GR? Why do you exactly say that the path integral is well-defined perturbatively? What does it mean other than a wrong statement? Do you mean just semiclassically?

    The scale factor has the "wrong" sign of the kinetic term regardless of the signature of spacetime that you choose. Why do you think that these things have anything to do with the signature?

    Concerning the time-dependent backgrounds, I would say - or agree - that it is difficult to Wick-rotate time-dependent backgrounds, and the known, easy physics in Minkowski space may be translated to a difficult physics question in the Euclidean space. In the HH state, it is manifestly the other way around. The Euclidean framework is more likely to give a meaningful answer.

    Further explanations from you welcome, Moshe!

    For the first anonymous after Moshe (maybe also Moshe?):

    We know how to analytically continue on worldsheets of trivial topology.

    But I really don't know how would I ever define the path integral over the worldsheets of Lorentzian signature. I don't even know how to classify the topology of Lorentzian Riemann surfaces.

    The only formalism in which the Lorentzian worldsheets are completely under control and can give you the S-matrix is the light cone gauge in which one can slice the worldsheets to slices with different values of light-like time.

    I admit that the "real" worldsheet are Lorentzian, but an easier and doable calculation is based on the Euclidean ones.

    Yes, I agree that the WDW equation acts on a state that describes "all times simultaneously".

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  32. The only reason people take the euclidean path integral seriously in some instances, is precisely b/c the analytic continuation makes sense *sometimes*. However the contexts are always limited, and invariably people have to check experiment or other methods to make sure it indeed works and/or matches.

    There are plenty of examples in 4d and higher where of course it does not.

    Now in this general cosmological setting where no complexification exists, it strikes me as rather unjustified. Further this ensemble of universes concept is troubling, we already know that quantization in 2d can start leading to incorrect results near the singularity with such schemes.

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  33. For the second anonymous after Moshe, i.e. the anonymous above Arun - are you Jorge? ;-)

    I suppose that you mean some proposals by Pullin and Gambini from 1998 and related stuff?

    In my opinion, all these "hopes" are manifestly impossible. Loop quantum gravity can't look like a conformal theory in the UV. It's claimed that they know exactly what it means - and most string theorists now know this stuff, too.

    A UV fixed point (do you really mean a *UV* fixed point, is not it a typo?) is a completely different possibility how the Planckian physics can look like, one that is not compatible with LQG which is discrete, i.e. very non-conformal, in the UV.

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  34. For the last anonymous in this discussion:

    Your opinions that you have outlined - and that you claim to be common sense - are not generally accepted. Ask Stephen Hawking. He would tell you that everything is about the Euclidean path integral, including the proof of non-existence of God. Even the Brief History of Time is full of the (exaggerated) statement that the Euclidean path integral is more fundamental. ;-)

    Hawking is definitely not the only one. A vast majority of the integrals relevant for relativistic quantum theories has been done in the Euclidean setup.

    The answer to the questions you try to address are mostly obvious. A theory must make sense so that there is any point in talking about the path integral. If it does exist and the background is time-independent, the Euclidean path integral is almost always the formalism of choice to make real calculations. Even for time-dependent backgrounds, the Wick rotation is usually a useful and often necessary tool - see how people tried to study the "S-branes", for example.

    Well, yes, I agree that the results of a particular calculation must be compared with experiments. This kind of discussion looks rather awkward to me. We're pretending that we're talking about a rather advanced and specialized technical question, but suddenly your comments jump to a thing as trivial as "results should be checked experimentally". Sounds kind of silly.

    Some required mathematical properties necessary for physical consistency (e.g. unitarity) can be proved for the Euclidean path integals, too, and it is often easier than for the Minkowski calculations.

    When you say that my statements (about the useful Euclidean path integrals) don't hold in higher dimensions, you're talking about something that does not really exist. Higher-dimensional quantum theories only make sense if they're a part of string theory. String theory has not been universally defined.

    Perturbative string theory is usually calculated in terms of a CFT and a Euclidean worldsheet path integral. AdS/CFT backgrounds are defined in terms of a conformal quantum field theory, and these theories also admit calculations in the Euclidean spacetime.

    The "possible problem" that you're trying to promote is not supported by a single existing example.

    I don't believe too much that you know some examples in which the behavior of Lorentzian path integrals (of a physically consistent theory) is less pathological near singularities than the Euclidean ones.

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  35. Hi Lubos,

    I think by now we agree about the facts, but only differ by our point of view. Let me then just clarify my last statement. I will be looking forward to catching you in person sometime.

    I was puzzled by your statement that HH prescription is the only truth in this game. In fact most my converstaion
    about the subject so far have been with skeptics...

    So, There could be two reasons for that statement - either that state is physically preferred (e.g by giving the right experimental results, by being very symmetric etc.), or it is unique.

    If I understood your viewpoint correctly, it is the latter, it is the unique state that makes sense in the formalism. To that I had two objections- first that it doesn't (it makes sense only semi-classically, at least that was the situation before Thursday). Secondly there are analogous situations where regularity in Euclidean signature picks a state out of many which are equally sensible physically.

    But anyhow, time will tell. It is certainly great progress to make the HH state mathematically well-defined, so one can ask the questions more meaningfully.

    best,


    Moshe

    nb: I think I always signed my comments, to really have those under my name it seems like I need an account or something, and I am just too lazy...

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  36. Heya Moshe,

    thanks for your clarification. Let me start with the following.

    Maybe I exaggerated my love for the HH state. A few years ago, I did not know what it was, and I would have probably told you that the question about the (preferred) initial conditions was not a scientific one.

    Today I will tell you that we don't know whether it is a meaningful scientific question, but if it is, the HH state is the most physically satisfactory answer for a preferred pure state that we have.

    "So, There could be two reasons for that statement - either that state is physically preferred (e.g by giving the right experimental results, by being very symmetric etc.), or it is unique."

    I don't think that either of these things you wrote is the reason why the HH state is conceivably the "right one". The HH state, in my view, simply seems to follow from the theory itself. We are testing the action and dynamics by other means - the evolution and experiments done billions years after the Big Bang.

    But when you apply the same action to the question about the initial conditions, I think that a fair answer is that it may give you the HH state, by the application of the very same path integral rules that we use for all other questions.

    Is not the fact that a statement/state follows from the right theory a sufficient reason to treat it a little bit more seriously than you do?

    I am not saying that the HH state is the unique state in the Hilbert space, even though you seem to think that this is my viewpoint - and I have no idea why you think so. There are obviously many states in the Hilbert space that make sense. ;-) I am just saying that the HH state is the state predicted by the theory as the initial condition for the wavefunction of the Universe.

    Let me make an analogy: thermodynamics. Imagine you want to study how a system of gases - or nuclei - at a certain moment of history when the temperature was high. We take exp(-H/T) where H is the Hamiltonian that we figured out differently, don't we? Would you doubt that exp(-H/T) is the right density matrix that desribes the Universe at temperature T, even though we can't test it for huge temperatures T? Would you call it a random, unjustified choice, that is moreover ugly because it is not "Minkowski" because it does not have the "i" in the exponent? ;-)

    In the same way, the HH state is likely to be THE right state that THE right theory predicts as the initial (pure) state. Is it really so confusing for us that we can't agree about it after 10 comments?

    Please accept my apologies but I still find your criticism of the HH state, even one in the last comment, illogical.

    You say that the "HH state is the unique state that makes sense". This is obviously not true. The HH state is just the state that is predicted by the theory to be the initial state of the Universe. There are many other states that are relevant for the Universe later, but that are not predicted to be the wavefunction of the early Universe. Then you negate your statement that the "HH state is the unique state that makes sense". This negation is OK.

    But before this negation, you also criticize that the HH state only makes sense semi-classically. That's a completely unfair, illogical criticism.

    Why is it unfair? It's much like in Feynman's criticism of the commercial for the Wesson oil ("it's not absorbed by the food"). Even though this statement may be "true", the fair truth is that "this oil is not absorbed if the temperature is low enough, much like for other oils, but it is, if the temperature is higher, much like for other oils".

    Why is your criticism similar to the Wesson commercial?

    The whole quantized general relativity only makes sense semiclassically; and beyond one loop, one must say more to define the theory properly. This is true for all aspects of quantum gravity, not just the HH state. So it's just not fair to pretend that this trivial feature - ill-definedness of GR beyond one loop - is an argument against the HH state. It's definitely not.

    Of course that beyond one loop, one must replace not only the semiclassical HH state, but the whole theory of gravity by a consistent completion (i.e. string theory), and the correct definition of the HH state is given by an analogous, more well-defined construction in string theory.

    Ooguri, Vafa, Verlinde have done it in a particular toy model example.

    Do you agree that you have not given any logical argument against the HH state?

    Best
    Lubos

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  37. Lubos,

    I think we can scratch my last two comments, as I obviously did not understand your point of view, so I ended up criticizing things you haven't said.


    But now I do not understand the very basic statement- if you find the correct evolution laws that predict results of experiments today, you would call it the correct theory. If its associated HH state is wrong for some reason, then you would say you have the right theory, with initial conditions different from those postulated by HH. Aren't those two questions, the evolution laws and the initial conditions, completely
    independent?

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  38. Lubos,

    On second reading, I think I understand what you are saying, no point trying to find wordings that we will both agree on, we are describing the same set of facts. Similar to your comment about yourself, I think you exaggarate my dislike of the HH state...

    Incidentally, the entropic principle would have been a wonderful title. At least you got to use it yourself.


    best,

    Moshe

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  39. Wait, you mischaracterize what I am saying.

    The Euclidean path integral when its possible, is very useful and mathematically is much better behaved than its counterpart (b/c of the exponential damping). I know of no case where it is *less* easy to use than the Lorentzian.

    Whats NOT clear is the question of when is it allowable? Some people automatically see a field theory and wick rotate immediately. Fine and good when you have experiment to compare too, or at least another method where you can compare answers.

    However, there are plenty of cases where this is not allowed and actually gives WRONG answers. You can peruse the mathematicians archives for examples. Now, it usually takes a lot of effort to actually see this, and in general its hard to know when it does or does not make sense before studying it in depth.

    However, at this stage of development, the Euclidean path integral to describe the entire universe (and presumably its ensembles) strikes me as religion.

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  40. I thought that the previous mixing of the metaphysical questions "whether we should check theories experimentally" with the technical question "should we use the Wick rotation" was just a typo.

    But you seem to be dead serious.

    These are completely separated questions. There is absolutely no rational correlation between the question whether our calculations should involve the Wick rotation on one side, and the question whether our theories should be tested experimentally on the other side.

    If you think that there is, you're making an error in your thinking.

    The Wick rotation is a standard tool to make any calculations - of Green's functions; S-matrix amplitudes, and so forth - in any Lorentzian quantum field theory.

    The reasons are mathematical and technical. The integrals in the loop diagrams, for example, have divergences that are much more controllable in the Euclidean spacetime. There the integration over the loop momenta may be reorganized into polar coordinates in spacetime which makes it manifest that the regulator preserves the Euclidean "Lorentz" symmetry SO(d).

    This means that the results analytically continued back to the Minkowski space will have the usual Lorentz symmetry SO(d-1,1).

    It would be harder to deal with the diagrams and their divergences in the Minkowski space - the angular integrals over the "angles" in Minkowski space do not converge. It's just technically more complicated to impose the regulators that preserve the desired symmetries, such as the Lorentz symmetry, in the Minkowski space. It can be done, and the most straightforward definition what this regularization should look like is the requirement that it agrees with the analytical continuation of the usual calculations done in the Euclidean space.

    This procedure - continuing the theory to the Euclidean space, calculating the quantities there, and continuing back - is normally applied to any quantum field theory to find its physical predictions. These results are subsequently compared with experiments, and the incorrect theories are eliminated. But virtually all theories are conveniently treated by this continuation to the Euclidean space and back. At the quantum (loop) level, the Euclidean language becomes increasingly superior, and at the non-perturbative level (think about the instantons), the Euclidean language is almost necessary.

    I don't believe that the people who claimed that the prescription of the Wick rotation gives a wrong answer have solved their task correctly. Or perhaps, they may have worked in the context of an inconsistent theory. A great example is loop quantum gravity. It's a framework that does not satisfy the standard physical consistency rules - the separability of the Hilbert space i.e. the existence of a countable basis, especially one that would resemble a Fock space at long distances; unitarity; Lorentz symmetry in the regimes of supressed curvature; and other requirements.

    Of course, if you start with an inconsistent theory, you can derive thousands of different results depending on the details of the calculation. The Wick rotation changes the results, and moreover requires one to make another infinite set of choices how to define the theory, and so forth. In LQG, one undoubtedly faces all these questions whether the Wick rotation is OK or not.

    But this does not happen in a consistent theory, i.e. a theory that is worth a physicist's time. The applicability of the usual calculational tricks that involve the analytical continuation back and forth is an indirect litmus test to see whether a given theory may be consistent.

    The Wick rotation is obviously OK - and the disbeliefs and doubts irrational - in a quantum field theory in the Euclidean spacetime. But it's also inherited by string theory in the Minkowski space (and of course, it becomes more subtle for time-dependent backgrounds). In string theory, we also have various instantons, including purely stringy instantons called D-instantons which are completely localized in the Euclidean spacetime and contribute to various processes - I mean real processes that can also be seen in the Minkowski space.

    It's more subtle to talk about the D-instantons in the Minkowski space.

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