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Distance between quantum field theories

The first and most interesting hep-th paper today is

Relative Entropy and Proximity of Quantum Field Theories
by Vijay Balasubramanian, Jonathan J. Heckman, and Alexander Maloney. They use the notion of relative entropy\[

D_{KL} (p||q) = \int \dd\mu (z)\, p(z)\log \frac{p(z)}{q(z)}

\] that is well-known to folks in machine learning (Kaggle anyone?) and related fields to quantify how far two quantum field theories are, how much information you lose when you run from the ultraviolet to the infrared.

Their formalism generalizes the Zamolodčikov metric on the space of couplings – that special case is only applicable if the field theories are conformal. (It's funny to write about these matters because I know Vijay, Jonathan, Alex, as well as Saša Zamoldčikov very well – and yes, Saša liked the Czech spelling of his name when I suggested it to him LOL.)

These formulae may be used to quantify the amount of fine-tuning in field theories in a new way.

But the main reason why I think that this kind of research is important, and I am doing something similar as well, is that similar "measures" may be applied to the string landscape and may be used to define rules of vacuum selection that Nature may have used Herself.

In particular, for many years, I have believed in something that I sometimes called the "misanthropic principle". The vacua in the "regions" where vacua are dense – where the average distance to tne nearest neighbor is tiny – must be disfavored, along with all the "highly populous classes" of the vacua that the anthropic believers tend to favor. In other words, we should live in some of the vacua that are "most special" and "most separated" from all the neighbors, according to the appropriate measure.

It's been my belief that such a principle or "bias" is needed for the probability distribution (saying which vacua are likely as initial conditions for the evolution of the Universe) to be normalizable.

This principle, if true, could perhaps be derived from some rather natural mathematics, some Hartle-Hawking-like wave function on the stringy configuration space or something like that. The wave function would drop as you would move towards the "dense, highly populated regions" of the landscape – for similar reasons as the reasons why the harmonic oscillator ground state wave function decreases for large values of \(x\) or why the Boltzmann distribution decreases if the energy \(E\) increases.

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reader ny-ktahn said...

I went to a meet up not too long ago where this was discussed :)

reader lukelea said...

Leave it to Lubos to coin "the Misanthropic Principle!" Hope it pans out.

reader Luboš Motl said...

'Windowsless' doesn't quite mean that there is no ventilation of the air. Instead, there is ecuperation or what is the word. The air is going through some tubes so that the chemical composition is recycled but the heat is not lost.

reader Dilaton said...

Are the QFTs to be compared looked at at the same effective scale (point in RG time) but lying on different trajectories of the RG flow, or are they on the same trajectory but at different effective times?

reader ny-ktahn said...

I'm putting together a folder. I might as well just put it out there. It is full of errors I might not even know of and at this point I don't even care what files are up there but may be someone can fix one of them or something. There are errors.

reader obg said...

Hey, thank you for all of the great posts.

I don't mean to spam this blog and be off topic, but a recent paper has a link between physics and machine learning techniques that may be interesting to you: http://arxiv.org/abs/1410.3831

also, as you know, the kaggle winner used some "winner take all" techniques. The researchers on that specifc network type just had a paper: http://arxiv.org/abs/1410.1165 ... you do make a good point that the boosting algorithm performs just as well, but it is nice to think that eventually there could be qualitative improvements in what machine learning techniques can do...

and lastly, in full linker not thinker mode, http://arxiv.org/abs/1407.1123 ... machine learning algorithms on a grassman manifold. It would be nice if "cluster algebras" or any other advanced mathematical machinery could help with the techniques in that paper.

have a good day and thanks for the posts.

reader ny-ktahn said...

bro! I'm not competing with their work. What I am puting out there is just incomplete stuff that I've been looking at some are a few pages , others just a page. None are referenced yet but the work from which I am attempting to build up from is in the same folder, and so forth. They are not new ideas. I just think I should put them here. I think I have some tiny experimental stuff too. In the event that a co-authors appears on my paper, please them. I am have no evil intentions here. For some reason I can't rename some of the files or open them. I hope they are the right thing. I just like apples and it is the name of one of the folders, I could not change it. I will also upload the link to the texts I use below. As I said it is not complete. If anyone can do something about any of the ideas stencils that would be great. I really do not intend to hurt anyone. I recommend the files be downloaded first and edited slightly as I wrote some of them when my emotions had the better of me. The stencils again are not original, they are just ponders about other peoples work but I can't bear to hold them as I can't add unique things to them, correct them and post especially after the events of today.


The texts I typically use were first given to me by Samantha. please spare her.


I do have two closer to complete papers that are a bit more on the unique side but they are not here. I will think a bit more about it, and then see if I can do something about it tomorrow.

reader ny-ktahn said...

obg, and ***Lenny*** , we don't need to fight. Not just because you've more or less colonized 4/5 of NYC but I might not have as much ammo as I thought. The #web phenomenon was cool but bro come on, summoning hackers against me and putting my emails out there . . . like seriously. I put videos on my webpage and you take down any shot at studying Physics that I might have had. Anyways in case you have not seen my page it has some updates from I guess Zombie Einstein aka stinker extraordinaire aka cracked iphone. Oh darn , I think the words I am looking for regarding my shortcomings are "exuberance of youth", Anyways just showing people that I put down a few summaries. Like I said if I had something truly new and remarkable done, then I would have it in the arXiv but I don't have anything like that complete. Guys let's move forward. Its a smart choice for everyone.

reader Luboš Motl said...

Different trajectories - obviously, one wants to quantify the distance for pretty much generally different QFTs.

reader Tom said...

Your misanthropic principle sure has a lot of intuitive appeal, Lubos, as your analogy with the harmonic oscillator indicates. A question in my mind is whether string theorists have studied topologies on all the possible vacua. I would think defining a sub-basis for a topology on the vacua might just be fairly straightforward for thinkers such as yourself who have developed the necessary intuition. Looking at the Borel sigma algebras that the resulting topologies generate could lead to measures of some interest. If one were lucky and a separable metric space resulted, the class of Hausdorff measures could be very interesting as some kind of fractal geometry seems plausible to me.

reader Ralph Hartley said...

I'm not sure I understand. Isn't there supposed to be only a finite number of stable vacua? A distribution over a finite set is always normalizable. If there are countably many, I still don't see it. Of course there is no normalizable uniform distribution over the integers either. I guess you could say that any distribution has to disfavor sufficiently large numbers, but large integers aren't any closer together.

How is density of vacua analogous to energy?

reader Luboš Motl said...

Dear Ralph, first of all, this paper is about QFTs, not vacua of string theory, and the parameter spaces of QFTs are continuous and uncountable.

Second, concerning the misanthropic principle in string theory, the total number of stable vacua is surely infinite and countable not finite - AdS5 x S5 with different values of N are enough to show a countably infinite subset. The finiteness only appears if one imposes the condition of 4 large (larger than XY) dimensions in some way.

Third, right, it's exactly the same point as the non-existence of a uniform distribution over integers, so any distribution disfavors large integers, as you correctly say.

Fourth, large integers are not closer together using the naive metric but the very point of this paper - as well as my thoughts about similar matters - is that a naive distance isn't necessarily the right one relevant for physics. The distance of QFTs they consider - and I envision for the misanthropic principle - is very different from the naive one.

Fifth, the numerical density of vacua is analogous to energy because the energy of the harmonic oscillator is mostly high at places with a large density of microstates (large x or p), while those with the small energy are special.