Dear Dr. Mittleman,

Thanks for your excellent question. Also I hope that the proposal in our essay does not contradict Piero's (i.e. Dr. Nicolini's) and his co-worker's paper since it was Piero who got me thinking about self-similarity and who explained to me some issues with self-similarity and QM/QFT when I visited him last fall. We should have cited the paper you mention and let me recommend this as an interesting and important paper.

That being said I think there is no conflict between our proposal and Piero's paper. In this paper Piero is interested in the self-similarity of a fractal space-time path. The self-similarity we propose is more abstract in that it is a similarity between the forms of the higher order corrections to the action (i.e. I_i ~ I_0 or the ith correction to the action has the same form as the 0th action and alpha_i ~ alpha_1 or the ith coefficients are similar to the 1st coefficient).

However in the essay we do make some connection between this abstract self-similarity and the more usual self-similarity of a path -- as one goes to higher order corrections I_i and alpha_i one is going higher energies and *usually* this means shorter distances. But in QG this is not necessarily the case due to the "UV/IR connection" which is discussed very nicely by Susskind and Lindesay in reference [1] of our essay. Basically the UV/IR connection means that the standard idea "higher energy --> shorter distances" breaks down at some point when gravity enters the picture. The argument very briefly is that at first as one goes to higher energy scales one does probe shorter distances as standard QM implies. However at some very large energy the particles one collides are within each others event horizon and the result of the collision is a BH with some horizon radius. The horizon radius now sets the length scale one can probe. As one increases the energy the horizon grows and thus the length scale one can probe *increases*. Thus our self-similarity is to be taken in the sense of each higher correction probing some higher energy scale. Conventionally this means shorter distance scale in QG this is not necessarily the case due to the UV/IR connection of QG.

Sorry for the long reply and feel free to ask follow up questions/comments. And also thanks for reminding of of Piero's excellent paper which we should have cited.

Best, Doug

Dear Singleton, Vagenas and Zhu,

Superb paper! I just gave you guys a big shot in the arm with a top vote score. The expansion of the action S = S_0 sum_n γ_nħ^nS_0 leads to results similar to yours. This is also interesting for in string theory the action is an expansion of the form

S = sqrt(-g)(R α'R_{abcd}R^{abcd} O(α'^2)

where α' is the string parameter that is O(ħ). In the vacuum solution for the Schwarzschild black hole with R_{ab} ~ g_{ab} this will lead to an expansion similar to the one in equation 7 of this paper.

I have long been interested in the idea there are connections between string theory and LQG. This expansion, both by you and with string theory appear to assume a classical background. The connection to LQG though might remove some of the issues with background dependency of string theory. Also the AdS/CFT correspondence indicates that four dimensional spacetimes have embedded within them the same data as in 10 dimensional supergravity or superstring theory. This means that 4-dim quantum gravity with loops or knots should contain the same information as in superstring theory.

Cheers LC

    Hi,

    Thanks! We look forward to any comments/advice. We will as well try to read your essay. As I mention above I am moving and doing some traveling in a week so it may take some time. Best, Doug

    Hi,

    If I understand correctly you are worried about the spectrum of emitted Hawking radiation being purely thermal? As you correctly point out this is a problem and in fact is a key point in the information loss puzzle of BHs. The approach we use to Hawking radiation is the tunneling approach pioneering by Wilczek&Parikh, Srinvinsan&Padmanhban, Volovik, Berezin&Boyarsky&Neronov (this last group was actually the first to propose a detailed calculation of Hawking radiation as tunneling but since the paper was published in a good but not so read Russian journal it does not really get the credit it deserves).

    In the tunneling approach it is possible to take into account the back reaction of space-time geometry of the BH due to the energy loss due to Hawking radiation. The effect of this back reaction can be seen in the omega^2 term in our equation (13) in the essay. For a purely thermal spectrum one only has the omega term. What we show is that this back reaction along with the assumption of self-similarity of the quantum corrections leads to a resolution to the information loss puzzle. In our essay information/entropy is not lost but comes out in the non-thermal radiation.

    Best, Doug

    Dears Colleagues,

    Congrats for this beautiful Essay. It is very well written and complementary to my one. You solve the black hole information paradox by showing that the number of microstates between the initial and final states is the same and this implies unitary evolution. My approach finds directly the unitary evolution through a time dependent Schrödinger equation. This permits me to write down explicitly the final state like a pure state rather than a mixed one.

    If you are interested on my approach you can see my Essay here: Christian Corda

    Best wishes and good luck in the Contest,

    Ch.

      Hi,

      Thanks and feel free to ask questions. Also we will try to find to look at your essay and others who posted comments on our proposal. Best Doug

      Hi LC,

      Many thanks for reading our essay and your nice comments. I will certainly try to read your essay soon. As I mentioned I will be moving in a weak (leaving my temporary position at ITB) but should have time after this.

      In regard to your comments you are correct -- our proposal still has the feature of background dependence, which as you point out is also a drawback of string theory. LQG might have some advantage in this regard but I have not followed the developments here closely enough to say anything useful -- but in the LQG talks I have seen the speakers always mention background independence as a good feature of LQG.

      Best, Doug

      Hi Dr. Corda,

      Thanks for taking a look at our essay and your comments. I agree that from reading the title and abstract it appears we are dealing with similar issues in a complementary way. There has been actually a lot of recent interest in the BH information loss puzzle. We will certainly have a look at your essay and send any useful comments/questions. As I mention above I am moving this week from ITB so it may be after one week.

      In any case thanks for the comments and reading our essay and best of luck likewise. Doug

      Please excuse me Professor Singleton, Professor Vagenas and Dr. Zhu,

      I am a decrepit old realist. I do not mean to be critical of your technically perfect essay; I would just like to comment on it.

      "In this essay we look at the connection between physical objects, i.e. "its" and information/entropy i.e. "bits," in the context of black hole physics."

      As I have pointed out in my essay BITTERS, one real unique universe can only be eternally occurring, once. Real unique, once cannot be connected. Every real "it" is unique, once. What you appear to be doing in your essay is simply finding an abstract connection between abstract objects and abstract information/entropy in the context of abstract black hole physics.

        Can we Wheeler a black hole?

        Is the nothing inside of a black hole real? No

        Is the something outside of a black hole real? Yes.

        Energy is made of particles and waves. What are black holes made of?

        Good luck, Joe

        If given the time and the wits to evaluate over 120 more entries, I have a month to try. My seemingly whimsical title, "It's good to be the king," is serious about our subject.

        Jim

        9 days later

        Hello Doug,

        Thanks for your reply; you don't need to apologize for its length! Sorry for not replying earlier; I've been distracted by various extraneous issues.

        At any rate, there's one thing that still puzzles me. I grant your point about the horizon radius increasing with the energy scale. What I'm not sure about is whether this point is applicable to the case of an evaporating black hole, in which the black hole's mass M goes to zero. For, as the horizon radius grows, it would seem that M increases, so that we're not dealing with an evaporating black hole at all. Or, if evaporation does occur here, it (arguably) happens "all at once" in a sudden burst of radiation that reflects instability of the black hole at (or below?) the Planck scale; and this doesn't seem to fit your model of an evaporation process characterized by self-similarity. Such an instability leading to evaporation is mentioned on p. 4 of Spallucci and Ansoldi's "Regular black holes in UV self-complete quantum gravity" (arXiv:1101.2760), in which the authors note the positive correlation between horizon size and energy scale. They address the above instability by arguing that the Planck scale represents a minimal size for black holes, which are stable with respect to this scale. In other words, they conclude that the horizon/energy correlation is associated with stable black holes of minimal size, rather than with evaporating black holes -- which brings us back to my question above concerning the (im)possibility of using this correlation in the case of evaporating black holes.

        I apologize if I'm missing something here, or if I've misunderstood your ideas.

        Thanks again for your reply.

        -Willard

        Hi Willard,

        Another good question. Actually Euro and Ansoldi's paper has some of the same features as our paper as well as an important difference which I will try to touch on. First if one looks at figure 1 of Euro's paper this is exactly the same point I was making about the UV/IR correspondence in my earlier reply. The hyperbola in the figure is the usual inverse relationship between energy/mass and length i.e. length ~ (energy/mass)^{-1}. The straight line curve is the gravity/BH relationship where length (horizon radius) goes linearly the mass/energy. The point where the curves intersect is the Planck mass/energy. Thus at energy scales higher than this scale one does not probe shorter distances any more but only makes a BH with ever larger horizon.

        Now the series we propose based on self-similarity should apply to an evaporating black hole. The higher terms in the series represent higher energy of quantum fluctuations and are not directly tied to the mass of the black hole (actually there is the following connection: in the standard picture of BH evaporation as the black hole mass decreases the quantum fluctuations become larger/more important). Thus our series should apply to the case of an evaporating BH since the higher terms in the series represent higher energy quantum fluctuations not (directly) greater or smaller mass of the black hole.

        Now in Euro's paper mentioned above and also in Euro's work with Piero they introduce a new feature that we do not have and this is where the difference arises. Euro and Piero have been working with non-commutative (NC) geometry - the idea that there is a non-trivial commutator between coordinates like x and y or x and z in the same way that in standard QM there is a non-trivial commutator between x and p. This NC geometry has the effect of introducing a minimal length scale into the theory (this is the theta parameter in Euro's paper). This has the very interesting feature that the IR/UV connection I mentioned is altered. This can be seen in figure 2 of Euro and Ansoldi's paper - they find that there is some minimal mass for a black hole. Lower than this mass one has a particle like lump or "remnant". Also in Euro's work with Piero they find that as the BH evaporates the temperature starts to decreases at some point and then goes to zero even when the object has a finite mass. They call this the "scram" phase since the black hole has been turned off in a manner similar to which nuclear power plants are shut down. Thus because of the NC geometry they work with the end stage of BH evaporation is different from ours where we would have the BH completely evaporate away. This was the reason we put "a conservative solution to the information paradox" since we do not assume any UV completion or cut-off or new UV physics. And let me say this is probably a negative feature for us since probably there is some new UV physics (a la NC geometry, string theory, loop QG) that comes into play. In fact I used Euro and Piero's "scram" mechanism is another recent work of mine on a new mechanism for inflation.

        So which do I "believe" - that there is some new UV physics or that something like the "conservative" approach we use in our essay is correct? Well no one really has a good idea of what goes on at the Planck scale so the best thing is to look at different options and hope/see if there are some low energy observations one can make which would indicate experimentally which path is right. Also as it turns out there is still one free parameter in our essay, alpha_1, the first quantum correction coefficient which we cannot/do not fix. Taking alpha_1 as a free parameter leads to the possibility, even in our case, of having a remnant as in the case of Euro and Piero (although our mechanism for having a remnant is different than theirs).

        Again a long reply so feel free to ask for clarifications or additional questions/comments.

        Best,

        Doug

        Hi Doug,

        Thanks for your helpful comments. I think my understanding of the Spallucci/Ansoldi paper is somewhat different than yours. It seems to me that their Figure 1 is meant to show that the Planck length is a minimal length that provides a UV cutoff: see p.3 of their paper, near equation 1. This cutoff means that increasing the energy of our probes beyond the Planck energy does not reveal anything new, i.e. it does not take us further and further into the UV, since the deep-UV is shielded from us by the event horizon of a Planck-scale black hole (this "shielding" is mentioned just above Fig. 1). As a result, we cannot obtain (trans-Planckian) higher-order corrections to the action by continuing to increase the energy of our probes; and so, it would seem, the self-similarity you describe gets broken at the Planck scale, at least on Spallucci & Ansoldi's account (and even apart from NC gemoetry).

        Of course, as you say, self-similarity can still be preserved by eschewing a UV cutoff altogether; but the (not implausible) possibility that quantum fluctuations get more and more unpredictable and out of control as we move deeper into the UV creates some doubts about such self-similarity. Nonetheless, as you also note, if your model can be connected with low-energy observations and supported by experimental results, then those doubts can be erased. So, I guess we'll just have to wait on experimental and observational progress in order to figure out what's really going on at the Planck scale.

        Good luck, and best wishes,

        Willard

        Doug et al,

        Every time I hear that string theory is background dependent, I want to reach for my gun. :-) I think that even by Lee Smolin's criteria -- and *especially* applying the principle of self-similarity that you have so elegantly incorporated into your thesis, string theory forms a complete quantum field of spacetime that is self organizing and self annihilating.

        I agree with Lawrence that your essay's position in the ratings does not at all reflect the quality of the work. I hope my high score to come will help.

        And I hope you find interesting my own essay that agrees with your conjecture of infinitely self similar curves. Of course, I'm sure you've seen Professor Corda's take on unitary evolution of pure quantum states, with which I also agree.

        (Lawrence, I'll hopefully get to your essay soon, as well.)

        All best,

        Tom

        Hi Tom,

        Thanks for reading our essay and as well reminding me that I promised read some essays starting with Lawrence's. I did download and read his essays and liked it very much so I'm heading over there to rate it now. Prof. Corda's essay is very close to our own and as well I had promised to read this as well -- it is downloaded but still needs reading :-(. Also if I understand correctly your essay has some connection with self-similarity so I should read this as well. Again thanks for your comments are reading our essay.

        Best,

        Doug

        Doug, et al,

        This is a copy of my respond to Doug Singleton on my essay blog.

        I have read several papers by Vladimir Dzhunshaliev on octonion field theory, and Merab Gogberashvili is a familiar name as well. Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of course classical mechanics is R. Gauge theory can be written according to quaternions H. A lot of gauge theory is done though in standard vector form without quaternions. It is interesting though that Maxwell formulated electromagnetism, the first gauge field theory, in quaternions. Field operators in a second quantization act on a Fock space basis to give quantum amplitudes. So we have a relationship that might be heuristically written as π:H --- > C. The question is then whether there is some sort of higher level structure π:O --- > H.

        Spacetime I think offers a clue. A black hole horizon has some quantum uncertainty on a scale near the string or Planck length. There will then be an associative uncertainty with three quantum fields, where one of those fields is identified near the horizon. The standard approach to QFT is to assign a harmonic oscillator at every point in space, impose equal time commutators on that spatial surface with the Wightman criterion for commutation, and work from there. Yet that spatial surface on a small scale will have some noncommutative structure and this will lead to a host of uncertainties in assigning QFT operators. If there are event horizons this should lead to an associative uncertainty.

        The above "maps" between C, H and O, where a similar map π:C --- > R would be the relationship between quantum mechanics and classical mechanics, are really just forms of the Hopf fibration. The relationship between quantum and classical mechanics is of course a difficult subject in its own right. With each of these "ladders" on the Hopf fibration there is some increased uncertainty. Quantum mechanics saved physics from the UV divergence that classical mechanics predicted with the hydrogen atom. Similarly this may protect physics from divergences with black holes, such as the singularity and maybe with the current big problem of firewalls.

        Thanks for the good word. I had a computer crash (virus attack etc) that erased my voting code. I also had it written down on a paper that also went missing. I have not been able to vote on papers for about a week. The FQXi people have so far not serviced my request that it be retransmitted. I have also been a bit slow in reading papers this contest cycle. I see that you have a paper in the list. I seem to remember that last year your paper was riding fairly high, where mine in contrast tanked.

        Cheers LC

        Dear Douglas Elias & Tao,

        I enjoyed your essay and agree that "energy conservation and the tunneling picture of black hole radiation allow us to show how the original "bits" of black hole information encoded in the horizon were transformed into the "its" of the outgoing correlated Hawking photons, thus providing a potential all orders in ~ solution to the black hole information loss puzzle".

        It makes perfect sense and was were I was going with my essay, but omitted some further detail which agrees with this. It would have taken another 10 pages in my case to add this aspect.

        However, part of the process is apparent when we note that both sides of 0 of the Fibonacci sequence, 2 remains positive. My representation of how Hawking Radiation releases energy is the -1, such that information sits evenly at the horizon and is maintained there. But also in 1-dimensional string like structures that may become Hawking radiation.

        Anyway please take a look, and I'm delighted to see such a fantastic essay doesn't contradict mine!

        Best wishes and kind regards,

        Antony

          Hi Anthony,

          Thanks for the kind words and we will try to have a look at your essay since your description sounds like there may be some connection. Since you mention "...1-dimensional string like structures that may become Hawking radiation." you may be interested in the work of Samir Mathur. He has a host of papers which treat the microscopic degrees of freedom near the horizon as string derived states which he calls "fuzzballs". I do not fully understand Mathur's approach but it is interesting a has the good feature of giving explicit microstates for the black hole.

          Best regards,

          Doug