## Wednesday, December 27, 2006

### Knotted strings, testeable string theory?

I have been keeping studiying the Clifford V. Jonhson book (D-Branes) and I gues I hvae a good understanding of the first six chapters. Specially I think I, at last, begin to understand the subleties of D-brane theory.

I have made a bit of crosscheckingof my understanding with the first chapters on string theory of the book of Tomás Ortin "gravity and strings" of which I also read the posterior chapters "extended objects". To complete my cross reading (and geting a good idea of what I´ll read next) I have re-readed in good detail the (year 2000) review article of B.E Baaquie and L.C Kwek "superstrings, Gauge fields and Black holes" (arxiv:hep-th/0002165).

As it is well known the Jonshon books is intended as a book in d-branes and not in string theory, and assumes some previous knowledge of them. I had it (that is a good thing because i´ll go fast in the next few chapters about superstrings). Even so I have reestudied some aspects of "basic" string theory. Mianly the Polyakov integral, conformald fields, and the theory of Rieman surfaces. I had studied it mainly in the Lüst-Theisen book (and partially in the Kaku book).

Now I have used the chapters of theBrian Hatfield Book (quantum field thery of point particle and strings) wich has a whole chpater about Rieman surfaces in wchich he dedicates a half part oof it to introduce in a mathematically correct formalism the concepts neccesary for a good knowledge of Rieman surfaces (begining from basic things as what is a topoly , a minifold, a complex structere and so that). Later it has a chapter in the pPOlyakov integral and a third one in vertex operators.

Well, apaart of the always goo habbit of recording things, why ree-study such basic perturbative strings? Afther all d-branes,T-duality and S-duality are, to a certain stent, a mean to study non-perturbative string.

Well, you can guess the answer from the previous post. I have been triying to formalice a bit that idea of knotted strings. Beeing a concept tied to "elementary" string theory my lack of a perfect knowledge of advance topics on branes seems, at least for the basic consideratios, irrelevant. Whay did I get?

As far as I see (I´ll still think it more in deepd and sutudy better the relevant questions) there is no inconsistency in thinking that strings could knot betwen themselves. I still am clarifiying which diagramtic in perturbation theory could yield such knotted state but still without that I can avance a few.

The great part is to refelxionate about what such a knotted state could mean. Closed (bosonic and "fermionic" i.e. supersimmetric) strings, the ones wchich can knott, at least if we are not thinking about how open strings could knot when they are tied to a d-brane, have in their spectrum the well known graviton, antisymmetric tensor and graviton (a supergravity multiplete for the superstring). Well, a graviton knoting around other is curious idea but it is not too promising to be an observable quantity in a near future, do you agree? ;-).

The fun begins when we compactify. For the begining we have the basic compactification of a single dimension which leas as to the usual history of T-duality, and the branes. But appart of it we have the "old" kaluza Klein mechanism. From it we hve that in the spectrum of the compactified theory there is a graviton in the reduced dimesnsions and, in adition, gauge vectors. What are that vectors?

We begin withf left moving 26 dimensional, closed, bosonic string and right moving, closed, superstring, we compactify the 16 dimensions of the bosonic string that we need to reduce to 10 dimensions in a self-dual lattices you have, yeah, you know it, the heterotic strings ;-). The allowed groups asociated to that lattices are SO(32) and E8xE8. Both of that groups contain the well known SU(3)xSU82)xU(1) of the standard model. That is why people had so good hopes for the heterotic string. What is the exact spectrum of the heterotic string?

(i) The ten dimensional graviton, antisymmetric tensor and dilaton
(ii) Their supersimmetric partners (gravitino and dilatino)
(3) These are the interesting ones for the current ppurpose, the gauge bosons of E8xE8 (or SO(32)

(iv) Supsersymmetric partners of the guage bossons, i.e. gauginos.

So, and these is the fun we have closed stirngs, which so allow knotting, who have in their spectra gauge bosons. Of course that gauge bosons are not the ones on the standard model. It is neccesary an aditional compactification to 4 dimensions in something like a Calabi-yau or an orbifold.

Or maybe other mechanisms as the warped scenaries of Randal-Sundrum in whhich you have extended, but only ascesible to gravity, dimensions. The 4-d world would be a d-brane to which open strings would be tied and the extra bulk would be only transitable to open strings, i.e. gravitons. If you have readed carefuly from these it seems that the gauge fields would also be open strings. I don´t know why these is so, because it contradicts my previous assets about the heterotic string. Obviously I need to study that things carefully.

But forgeting by now that remianing questions let´s state, at last, the gfreat rsult. If you allow knoted strings these represent particles which must propagated "together". FOr all practical purposes you couldn´t distinguish from scatering experiments (suposedlly the only ones allowed) unknoteed and knoted states. But if you have a knoted state of, lets say as an example, two gluons the "resulting" state would be diferent in ther properties of any possible single gluon. The same would work (with care to the higgs mechanism) for W bosons, the responsible for the elektroweak force, you know.

So you could have some composed state of W bossons wich would be diferent of a single W boson. And as far as W bosons are observbles that would mean that if knoted states exist they predict what, for any practical purpose, would llok as a new kind of particle, which, from an stringy perspective would be a composed state.

Why am i publishing these in these, semengly totally unknown, blog and not in a good review (such as nuclear physics B) or at least in Arxiv. It is obvious, there are a lot of technical aspects to be checked. But in prevention of someone else having the same idea and pubising it in a more precise way i left here these preliminar version as a proof that at least partially I was the first one (asfar as I know) to have the idea ;-).

Of course I will have made some stupid mistake and it will be mostlly a proof of my missunderstanding of the theory, but anyway here it is :-).

P.S. The idea of knotting strings has a few subleties. For example, it is not totally stated in the intuitive picture of the theory what happens when two strings are bringed together. The ony picture is the well known "pant diagram". The existence of knotted states would require that, at least to some stage, two strings couldn´t cross each other (something coherent with the pant diagram). These would mena that once formed the knoted state couldn´t break when an string simply cross each other. theformation of the knoted stated would require some mechanism, maybe implying itemediary open string states.

P.S. 2. In the observational side there is an even more interesting posibitie. If for some reason a closed string couldn´t easilly go around the d-brane where an open string is tied that would mean that open strings could somewhat tied up gravitons. That could result in some kind of gravity shielding. Also it could mean that ther could be unexpecteds relationships betwen gravity and elecromagnetism. It would be interesting to examinate from these viewpoint the controversal experiments of podkeltnov and a few other relted to gravitomagnetism.

P.S. 3 Too much speculation going here. Yeah, sure. I´ll try to go into more precise statements, but it I cant´go too further in that purpose at least I have a very good speculative ideas to include in science fiction writngs, which is another thing I like to do. In fact string theory is a lot better that LQG for writing science fiction. It is an open question (at least for me, even beeing aware of the interestings drawbacks Lubos Motl usually post about LQG) that it is so much better than LQG to do real physics, that´s why I try to learn both of them.

## Saturday, December 23, 2006

It has been a few weeks that i dónt publish anything. I have been learning a few buch of things while.

I had posted in these blog some doubts about the foundations of string theory. I also posted some of them in physics forums. If you don´t want to push the link I´ll give you some of the answers I got:

Dmystifier:

here is no such thing as constituent points. A string can decay or snap only into other strings, and the lowest energy configuration is going to be stable. See some other recent thread here on a similar issue.
..............

R.X.

there are simply no constituent "points" on a string. Namely how could one possibly ever measure or see those? One would need to do a scattering experiment and bounce something off that string. But all what one can do is to take another string and use it "as a probe", ie, scatter it against the given string; what would come out from this experiment would be just other strings, because the only interaction that exists is splitting and joining of strings. This is related, as you say, to the notion of a minimal length scale beyond which one just cannot see. Thus, "points" on a string are not observable and thus, by the rules of quantum mechanics, are meaningless quantities.

One should not literally think about strings as little filaments made of "something else" - they are quantum mechanical oscillators and in order to understand them, one should not use too a naive classical intuition.

When he says that I mention it he refers that I had talked about T-duality . What is T-duality? Or better, what are dualities at all?

Well, dualities are symmetries betwen strings theories in diferents backgrounds or in betwen diferent string theories (or even betwen string theories and other theories)

The most widelly stuided, and may be the most important for the actual development of string theory, is T-duality. If you compactify some of the extra dimensions of a closed string theory in a circle of radious R you have in adition to the usual discretization of moment, propious of point particle physics a purely string efffect. It consist in that the closed string can wind around the circle a certain number of times usually denoted as w.

Well, the key point is that the observables of the theory (mass, scaterging amplitudes, etc) are invariante under the combined exchange:

$R \leftrightarrow \alpha/R$

$n\leftrightarrow w$

There exist also T-duality for open strings. That duality is one of the ways D-branes make their aparition in string theory. And once you have the D-branes you can make some kind of T-duality among D-Branes (branes also can twist around compactified dimensions xD), but I will not extend myself in that questions. Only to mention that the entry (of today) of wikipedia in these topic mention that T-duality relates type II-A and type II-B superstring theories and that mixes betwen them the two heterotic strings. Right, but it is easier to study T-duality for the bosonic string to begin with the topic ;-).

Afther explainingg the T-duality to explian how it relates to the problem we have now. Well, the important part is the $R \leftrightarrow \alpha/R$. these means that you can´t distinguish distances smaller that the radious of compactification because if you try to go there it is as if you would go to a greater radious.

These is the compactification radious, presumibly of the order of the planck size for usual scenaries of compactification, and not any characteristic legth of the string. So my claim in that post in physic forums was a bit diferent to the R.X. answer who addres the imposibility of seeing points to string themselves and not to the compactification.

Afther having discused these and considering self-explanatories the answers of demistifyer and R.X., what is my viewpoint about the dispersion of strings under evolution? The reader can judge himself. Myself I find the allegations, specially the last one of R.X. interesting and I´ll think about it. Anyway if the natural interpretation of the math is naive i guess it could be interesting to make a somewhat different formalism in wich that interpretation couldn´t appear. Maybe something as talking a bout a rule for an equivalence kind of points and the reasons why you that equivalence. Afther that you could explain that a very natural realzation of that equivalence class can be viewed as a mathemathical string. Of course these is just a very personal viewpoint, and one wich needs a further development.

To end these entry I reproduce here a diferent question about string theorie which I explain in that thread:

Investigating about a (very) older theory about extended objects, the knot theory of tompshon, tein, Maxwell (partially) and others inthe XIX century I discovered they had a very reasonable argument (withing the context of their knowledge of physics) for considering them. It came for a theorem in fluids mechanics with stated that once formed a vortex in a perfect fluid It would remain stable forever. In their times it was assumed that there was an universal prfect fluid, the ehter. But, of course, once the ehter theory was discarded the theory loosed any support (and Q.M appeared as a much better theory for the microscopial physic). Of course people who belive even nowadays in some kind of ether could claim for an string theory as vortex of that ether (well, maybe), but certainly mainstream string theory physicist hate ether (with good reasons, IMHO).

Maybe if there would be a way to see an string as a solitonic state of somtehing else I could see areason for an (at least partial) stability for them

By the way, in that times the tried to explain spectroscopic results as knotting of two or more vortex. That raised me a new question about string theory. Why strings can´t not knott around themselves?

I mean, if you would accept (as everybody does) that strings are (clasically) stable beeing quantum objects ther would be the possiblity of a closed string could be created in a knotted configuration with another closed string.

And a last question. These is about the polyakov integral and the admited interaction vertex (not confuse with vertex operators). It is allways showed that you can see an split of an string in another two, but, whay about a vertex in wich an string splits in thre, four, or in general N strings? What forbides the existance of that vertex?. I admit that perturbative theory with, vertex operators, dhem twists,moduly and teichmuller spaces is something wich I have readed a few times but I still don´t fullly understand. But towards my understanding works I don´t see a good reason for multisplitng vertex (or "knotting" vertex if we accept going from Rieman surfaces fto more general complex, algebraic curves with some singular points).

B.T.W. I mentioned in a past entry that LQG, had scenaries in which from "only gravity" the made to appear point particles (and may be even strings). Well, althoughtnot in deep but I readed some of the papers and I have a general vision of their arguments.

On one hand there is the Smollin-Markopoullous-Billson Thmpson paper. It is formulated in the framework of canonical (or hamiltonian) quantum gravity and it is based on preoon models. Beeing based on canonical L.Q.G it has no dynamics (because the hamiltoninan of LQG is a constraint, that is null, so it can´t give any evolutions, at leas in a conventinal way. That also true for the, easier, ADM hamiltonian of gravity).

On the other hand is the Baratin-Freidel model. It is based on spin-foams version of LQG (you can so it as the "lagrangian" version) has dynamics. They argue that from an scenary of pure gravity they can reproduce the Feyman diagram of any point particle (or even maybe of an string). They did it first for 2+1 gravity and recentlly for 3+1 gravity. That´s their claim. But as far as i have seen they introduce by hand the feyman diagram, rewrited in the spin-foam technologie so it doesnt appear in a dynamic scenary form pure gravity. Anyway I need to read it in more detaill so don´t trust these preliminary drawback as definitive.

And for now that´s all folks.

P.S. I hate these stupid scripts who try to correct the html synthax. They don´t like pure html and they try to convert it into XHTML. In the proccess they try to correct things as spaces or non asccii elements in the source etiquete of an image tag. But if that img tag is LaTeX code for a public LaTeX server that can corrupt the code and the images are not seen. But it is even worst. It try to obligate you to use XHTML but the page itself is not XTHML, and you have not access to the head etiquetes (or at least not in any reasonably easy way) so you cant make a doctype declaration wich would allow you to use MathML, wich would be an alternative to Latex. I´ll try to correctlly publish the latex images if there is sme way to prevent the self correction of html, but i am not sure if it will be possible today. If so it could be you don´t see the images (formulaes) correctly