Dear Peter,
I thank you for a very useful contribution to an approach that needed to be explored. I am one of those who prefers to experiment with deviation from Lorentz covariance, but knowing some of the consequences and possibilities of changing other hypotheses is quite valuable.
Good luck with the contest, and take care,
Ben Dribus
Peter,
Your essay is far better than the middling status it has in the community rating. I have given it a basic read through, though I intend to give it more focused attention tomorrow. When I am done here I will give your essay a high score to bump it up the ranking.
You raise some interesting issues here. You write
Δ(x-z) = i(G_r(x,z) - G_a(x,z)) = [φ(x), φ(z)]
which time ordering is behind the commutivity. In my essay I argue on the basis of M-theory that field locality is to be removed. In effect [φ(x), φ(z)] =/= 0 even if x and z are spacelike. This is a departure from your work, but I am wondering if one can have a propagator equation
Δ(x-z) = i(G_r(x,y) - G_a(z,y)) = [φ(x), φ(z)]
Where x,y are timelike as are z, y. The equation 5
ξ(x) = T^†[e^{-iL}]φ(x)T[e^{-iL}]
would replace e^{-iL} with Jacobi theta functions.
I will probably have more comments directed more closely to your essay tomorrow.
Cheers LC
I doubt any bump will be enough, however the thought is welcome.
In your second equation here, I'm not sure what "y" is? There's something to bear in mind (at least within the axioms of AQFT), which is that if $\Delta(x-z)$ is a function only of the separation between the points identified by "x" and "z", then the field has to be a generalized free field (LICHT AL and TOLL JS 1961 Nuovo Cim. 21 346-51).
Do you mean by your last remark that you want a non-unitary evolution? I don't think you mean to replace e^{-iL} by something like [math]e^{-i \ln(\theta(...))}[/math]? What motivation do we have for adopting a Jacobi theta function in /this/ context?
I was conjecturing a Δ(x,z) formed from retarded and advanced potentials with a common point y to the future (timelike or lightlike separation) of both x and z. The propagation from x to y in the retarded case and from y and z in the advanced case is replaced by the commutator. This of course would need to be derived by different means of course. Yet quantum gravity implies a nonlocality of fields, and this seems like a possible way to derive this. The trick is to insure that no information or quantum bits are communicated between x and z. My conjecture here of course could be wrong.
The replacement with theta functions is a way of possibly treating black hole singularities. In that case the evolution equations contain a singularity and are meromorphic, rather than unitary. The theta function is
θ(z,t) = sum_{n=-∞}^∞exp(πin^2t 2πinz)
which takes the place of unitary functions such as e^{-iL}. I will not right now go into a complete discussion on this, but these modular functions I think in general replace unitary functions
One way to see a motivating reason for this is with the following metric with 1 - 2m/r = e^u. so then
ds^2 = e^udt^2 - e^{-u)dr^2 dΩ^2.
We now have to get dr from
dr = -2me^u/(1 - e^u)^2du.
Now the metric is
ds^2 = e^udt^2 -2m[e^u/(1 - e^u)^4]du^2 dΩ^2.
The singularity is at u = ∞, where the dt term blows up, and the horizon coordinate singularity at u = 0 is obvious in the du term. My rational was that the singularity had been removed "to infinity" in these coordinates. This makes the black hole metric similar to the Rindler wedge coordinates, which does not contain a singularity. In the accelerated frame or Rindler wedge there is singularity. The treatment of the Schwarzschild metric in the near horizon approximation Susskind uses is one where the singularity is sufficiently removed so that field in the Rindler wedge may be continued across the horizon without concerns. In this metric of mine the singularity is at infinity so the analytic functions for fields in the Rindler wedge are replaced with meromorphic functions with a pole at infinity.
The singularity at infinity causes trouble with the end point of the radiance process for it has to "move in" from infinity. The final quantum process of a black hole is a problem not well known in any coordinates. By the time the black hole is down to its last 10^4 or 10^3 Planck mass units the black hole itself is probably quantum mechanical. In my coordinates (assuming they are unique to me, which is not likely) the singularity at infinity may not have to "move" from infinity. There may be some nonlocal physics which causes its disappearance without having to move at all. This nonlocality is a correspondence between states interior to a black hole and those on the stretched horizon. The Susskind approach does not consider the interior, and he raises this as a question towards the end of his book "The Holographic Principle."
Unitarity is represented by a complex function e^{-iHt} and so forth, which is analytic. The loss of unitarity does not mean there is a complete loss of everything; in particular quantum information can still be conserved. A simple analytic function of this sort describes standard quantum physics. Gravity as we know is given by a hyperbolic group, such as SO(3, 1) ~ SL(2,C), where the latter has a map to SL(2,R)^2. The functions over these groups have posed difficulties for quantum gravity, for they are explicitly nonunitary. The trick of performing a Wick rotation on time or with τ = it is a way of recovering the compact groups we know in quantum physics.
It does turn out I think that we can think directly about quantum gravity by realizing that the SL(2,R) is related to a braid group with Z --- > B --- > PSL(2,Z), and that the braid group is contained in SL(2,R). Braid groups have correspondence with Yang-Baxter relations and quantum groups. The group SL(2,Z) is the linear fractional group, which is an elementary modular form. An elementary modular function is
f(z) = sum_{n=-∞}^{n=∞}c(n)e^{-2πi nz}
which in this case is a Fourier transform. In this case we are safely in the domain of standard QM and QFT. In general modular functions are meromorphic (analytic everywhere but infinity) and analytic condition is held on the upper half of the complex plane.
Of particular interest to me are the Eisenstein series of modular functions or forms. These define an integer partition function, which is an acceptable partition function or path integral for a stringy black hole. I include a graphic here illustrating an Eisenstein function. This has a certain self-similar structure to it, or what might be called an elementary form of a fractal. In this picture unitarity is replaced with modularity. In this more general setting the transformation do no promote a field through time by some operator, but that the operator simply computes the number of states or degrees of freedom in a way that is consistent. Unitarity is then a special case of this, which happens to fit into our standard ideas of causality.
The Eisenstein series describes a partition function or path integral for a black hole. The theory is not one of unitary evolution, but simply one of counting states or degrees of freedom on the horizon. In effect physics is more general than unitarity, where unitarity is a necessary condition to describe the semi-classical states in postulate #2.
Your essay went up by quite a bit when I scored it last night. I think it went back down some. To increase your community rating it is best to read other essays and leave commentaries there. You will of course find a whole host of essays that are nonsense. I try to down vote those to clear them from the upper ranks. Comments left on the blog site of other reasonable authors, with some attention drawn to your essay, can help raise your rating. If you "demolish" bad essays higher up it is of course best not to write anything. There is a bit of the "art of the schmooze" involved with this.
Cheers LC
Taking Lorentz covariance to be broken at small scales is of course a much pursued option. I'm not against such ideas, but there seem to be so /many/ possibilities, and not, I think, quite enough empirical justification for any of them. I can't figure out which approach I like enough to work with.
On the other hand, the kind of nonlinearity I introduce here is not an attempt to change interacting QFT, it's more an attempt to understand and make more accessible (albeit falteringly) the role that is played by renormalization, which I take to be an empirically successful mathematics (notwithstanding the reservations one might have about the quantum gravity regime, say, dark matter, etc.), without introducing the sophistication of generalized functional methods, which I take to be more inaccessible. I would hope that a more transparent mathematical structure and better understanding might lead to better engineering in future.
FWIW, I have just re-read a review article on renormalization, Physics Reports 511 (2012) 177-272, that motivates the mathematics almost entirely in terms of language that is very much that of signal processing, "the natural description of physics generally changes with the scale at which observations are made. Crudely speaking, this is no more high-minded a statement than saying that the world around us looks rather different when viewed through a microscope. More precisely, our parametrization of some system in terms of both the degrees of freedom and an action specifying how they interact generally change with scale." This could all be subsumed by a discussion in terms of different choices of window functions. Of course details are important, so that this kind of introductory discussion cannot be taken too much to heart, but it seems striking that all the observations we make are supposed to be made using the same type of microscope, whereas if we work in a formalism such as I propose, we might use a different window function, representing a different regularization for each observation. [I take it that this micro-addendum to my essay will be seen by almost no-one, and I realize it's not really a propos of the comment that has engendered it, but whatever.]
Reading through your essay, I think it's justifiably high in the community rating, and you have my best wishes, but I find myself nonetheless finding that the whole thing is not to my taste (which is too personal a feeling to leave as a comment on your essay, however). It's very much with apology, because it's somewhat classic that people who are drawn to algebraic QFT-type approaches have been left far behind the mainstream and have become quite marginal. Ah well.
Dear Peter,
I appreciate your response, and particularly your honesty. "Taste," (or intuition, as it might be called) is not to be dismissed out of hand. You may have noticed from my bio that I study algebraic geometry (mostly complex algebraic geometry), so nonmanifold models were not initially much to my taste either. The journey toward considering such things is not something one can put in a 12-page essay!
I think that your remarks about window functions are interesting, and I have great respect for approaches that make systematic use of scale. I studied harmonic analysis/wavelets a fair bit before going toward the algebraic side. Take care,
Ben
I've eventually succumbed to commenting briefly (and not substantively for the purposes of the essay contest) on your more-or-less equating "taste" and "intuition". I think taste has an additional component, that of intention. That is, I think of taste in Mathematics/Physics as more colored by what we wish to achieve, whereas I think of intuition as more directed to how to achieve what we wish.
This is somewhat comparable to some ways of defining rationality in Philosophy; Wikipedia offers, inter alia, "A rational decision is one that is not just reasoned, but is also optimal for achieving a goal or solving a problem." That is, for my point here, /what/ goal or problem we choose to address is more a question of ethics, something that is not open to reasoning in the same way as is the /way/ in which we address the goal or problem that we choose.
Of course such boundaries are quite fluid in practice. We might well persuade someone, for example, that their goal of the moment does not further their achieving a more ultimate goal.
Your comments, of course, have been thoughtful and helpful, here and in the other places that I have seen; please take this as just a by the way.
Best wishes, Peter.
Dear Peter,
Your essay is interesting for me. I think you can find some of answers of your questions here:
Best wishes
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
