Why Loop quantum gravity?

I had a few entries in this blog wondering about some unnaturallness that i find in string theory.

On the other side I am aware about what some people in the string theory comunity think about LQG, so I have decided to make a post about why someone could worry about LQG nowaday (or why not).

One important thing is your academic enviroment. One of my teachers (probably the best in the field of string theory, in my facoulty, when I was studiying) played atention to LQG in his general reviews about quantum gravity. By LQG I mean a broad conception which included to explain the wheeler-de Witt equation and the Astekhar connection in paers previous to the actual form of canonical LQG. In newer papers he stills includes some information about LQG (altought his main interest is in strig theory). I never asked him personally about his opinions so maybe he was just beeing polite, but from his papers one would think that it is not crazy to study LQG. So judging from direct influence of your teachers it was a free way to study LQG.

For a brief (unhappilly too brief) time I beguined to study for a tessis in geometrical quantization (in the math faculty). When later I saw that geometrical quantization played a role in LQG I really enjoyed it because it somewhat meaned that the time I studied geometrical quantization was not lost time.

As I stated before in this blog for a time my knowledge of string theory came mainly from the two books of Michio Kaku and the Lüst-Theisen (and secondary in theh Hatfield book "quantum field theouy of point particles and strings"). While they cover fine the "first revolution" string theory I must say that the second KAku book (string theory and M-theory) althought readed a posteriory when you allready understand the topics, is correct I feel that it doesn´t makes a good job presenting the intuitive ideas of the second string revolution. It would need more space to trate the topics.

Another aspect is that in my faculty library always there has been a democratic representation of string and LQG. You get all the new books in string theory (nowadys you can get the Michel Dine and Becker-Becker Schwartz ones) as well as books by LGG people (for example the book o Baez "lopps, knots and gauge fields). Also you can have books in euclidean quantum gravity and other topics (A hurray for my faculty library).

Also I must say that previously to going hard into string theory I wanted to have an as strong as possible basic in general relativity and quantum field theory. The quantum field theory books usually contained an intro to string theory (kaku´s book on QFT). Pariticularly usefull I found the book "particle physics and cosmology" to understand some more advanced topics in QFT and physicis beyond the standard model. But more important than QFT was for me general relativity. My main source where the book by wald and a really extense book about black holes writen by Frolov & Novikov. Also I found very usefull the online website "living reviews on relativity". I think that I don´t need to say that a general relativity formation favours to apreciate LQG over string theory.

Well, all these means that there is no a priory pressure agianst LQG. This leads me to a famous paper in 2003 (or around that date) in which it is presented the new (for that time) canonical LQG, with the very interesting promise of a soon to come experiment (the one related to the MAGIC experiment that I bloged in the last entry). As a complement living reviews had a somewhat complementary introduction to canonical LQG. You can doubt about the goodness of he physics behind, but as an introduction/presentation both articles (specially the one by Thieman) are ver well writen.

With all these background to study LQG and not string theoy seemed a very good idea. Also an a priory netural forum, physics forums, was very pro LQG (by that time the forum in www.superstringtheory.com had been hacked).

Well, what came next? I guest that LQG has somewhat killed itself. Canonical quantum gravity seemed a very elegant and usefull reformulation of the old idea behind wheeler de witt equation (reached from canonical gravity, not euclidean quantum gravity) But very soon the atention was deviated to spin-foams, arguably because of "the problem of time". But spin foams are not so obviously related to gravity as canonical LQG. I mean, there one starts not from the general relativty lagrangian, that is what one, at least naively would expect but from lagrangians which hopfully, with some constraints, reproduce classical general relativiy. Worse still, it is not clear which of all them is favoured so people study many of them (ok, the Crane one´s is the most favoured, seemingly).

But the history doesn´t end there. It is adviced to study causal triangulations, and quantum groups. Not to say there is not a clear relation betwen all these aporaches (besides a like ofr discretized space-time). I must say that I studied spin foams, mainlly thanks to the review articles by Alejandro Perez, which I find very well writen, and which, IMHO present the physics ideas a lot better than the reviews by Jon Baez. What I couldn´t find the interest enoguht to read are the recomended literature about causal triangulations and the like (maybe because I never liked too much lattice QCD, but I guess that it was not that the ultimate cause).

Later I found Lubos Motl blog and it´s comments about some famous papers in LQG (as for example the one in the calculation of the graviton propagator), and well, I beguined to be aware abouth that fraticide strings wars and the "not even wrong" blog (whcih at first I thought it was a blog pro string theory, why else would someone would devote so atention to string theory afhter all? :P).

Most interesting that his attacks to LQG I found interesting in Lubos Motl the presentation of some intuitive ideas of string theory that were somewhat ausent in the literature I had readed.

Well, nowadays I don´t really follow too much LQG. I am aware about it´s developments while I am devoting time to update my string theory knowledges (I feel I am ending that task at last, at least in the main fields, and some not so mianstream aspects also). Would I give any advise against LQG?

Well, LQG people, at least someones, have a like for "Filosofing". I find entertaining reading the prose of their papers (I specially recomend one containing a discusion about the entropy of black holes and if some would count or not the "inner" degrees of freedom. I would recomend people to read also the introductory papers that I have cited, and maybe also some paper of Router about his claim that conventinal quantum gravity has a renormalization group fixed point. Later I would recomend that people to read some of the frequent blog entries on Lubos Motl blog critizazing LQG approachs. At that point it would be an individual decision of every one to go further on LQG or not. But I think that not devoting the relativelly few time that it requires to read the papers I mention(if somone is interested I could give him the exact references if he is not abble to find them himself) is not a good idea.

Well, I have been almost a mounth without any post. In this time I have been reading about some topics, that I hope that my eventual readers could find interesting.

MAGIC experiment, spacetime foam and wormholes

The most relevant new of this month was, probably, the announcement of the MAGIC discovery of indications of the possibility that light speed could depend on its frequency. The relevant paper is this.

it has been discussed in some blogs (Lubos, Peter Woit, Sabine) and some forums (for example physics forums or, in Spanish, migui´s forum. See the links of these blog to search them if you are interested). I will not add too much about it, at least not still. Just to state that, if truth, it would, probably, be a signature of a quantum gravity effect, and that would be the first quantum gravity effect ever observed, which is, of course, a very, very important thing.

The reason why these could be a sign of quantum gravity is also an interesting thing. LQG people reach to that conclusion from two sides. The best known one comes from the side of canonical LQG. There they state that the area operator has a discrete spectrum on the kinematical Hilbert space and that is an strong suggestion (but not necessarily a definitive one) that there is a minimum length. They have tried to probe the same result directly for the length operator but until now they have obtained only preliminary results because of redundancies on the quantization procedure (or something like that). Also it has been criticised that in the full, dynamical, Hilbert space the discreteness of the area operator could fail. Of course Lubos Motl and the stringy community knew that these result of discreteness made no sense long time ago. Pity that I have not seen a careful mathematical proof of their wordy arguments. Well, anyway, once you have a minimal length you can make a DSR (double special relativity) theory and get modified dispersion relations for the propagation of particles and you get the desired result presumably observed. DSR theories have many problems and nowadays seem to have derived into something named ESR (extended special relativities) which in some way results into some kind of nonconmutative geometry.

Other way, worst known, in which LQG people arrive to this kind of results is from a certain limit of spin foams which results, agian, into an NCG, see for example these paper for the details.

It is interesting to note that in fact any generic NCG (non conmutative geometry)could, potentially, give rise to a breaking of Lorentz symmetry and, hence, to a frequency dependent speed of light. Critical string theory could result, through a non zero vev (vacuum expected value) of the NS-NS antisymmetric tensor give rise to an effective theory describable by an NCG. Seemingly these would mean that critical string theory could explain these result, but in view that Lubos Motl doesn’t point in these direction maybe I am missing something. Also it is courius to note that critical (super)string theory is formulated in ten dimensions to avoid both, the conformal anomaly and, related to it, the Lorentz invariance. Even thought a solution of it, the NCG limit, violates Lorentz symmetry. Theses is not new, the solutions of a theory need not to respect the symmetries of the lagrangian.

Well, these are ways in which some could achieve the MAGIC ideal result (if the alternative explanations with no new physics could be discarded). But the paper doesn’t rely on any of these. It is related to a very special kind of strings, the Liouville strings. I had a previous knowledge of theses, and also of the fact that the mainstream string community had not in good estimate the works of their mayor proponents, Nanopoulos, Mavramatos, and all. I had inquired some people of the string community about these Liouville strings when it was announced in the CERN courier that they predicted the vacuum frequency dispersion of speed treated in these post (B.T.W. it is important to note that at least in the CERN courier paper the clearly stated that LQG perditions and Liouville string predictions were in the opposite direction, I have not seen people being to precise about these concern now).

At that time, 2003. I preferred to study LQG, which looked very promising at that time, and I didn’t pay mayor attention to Liouvile strings, the treatment of then in the Hatfield book "quantum field theory of point particles and strings" was dissuasory enough. For someone who doesn’t know anything about Liouville strings a quick comment. You treat the conformal factor of the world-sheet metric as an independent field. In the end it results into a Lagrangian for it which resembles something related to something previously known as Liouville field (I don’t know the exact reason). The problem is that the lagrangian contains an exponential of the field so it is very hard to work with it. In the Hatfield book it is shown that these problem is related to the fact that the 2d bosonic fields of the world-sheet don’t decouple from the 2d gravity and that means that you are doing 2D gravity. It was studied through matrix models (not the same matrix models related to M theory) and at least in the epoch of the Hatfield book they didn’t allow a reasonable formulations in D<25, that is, they didn’t allow a formulation of string theory in four dimensions which was the main goal of that theories. Probably later the have had some improvements, still i am not too sure.

My idea, when I first tried to think how Liouville strings could result into the MAGIC effect was very obvious; they brooked Lorentz invariance, which was enough. But I begun to read the papers linked in the announcement article and I got shocked. They talked about decoherence, hawking radiation, the role of the Liouville field as an emergent time and the role of spacetime foam as the source of the effect. Separately any of these statements seem bizarre, but together, well, at last I understood why Lubos didn’t even mention "liouville strings" in his entries about these announcement ;-).

I didn´t read in deep any of the papers of nanopoulos and all, but I have made some partial readings related to some of the aspects. I’ll speak a bit about the spacetime foam (not to confuse with the LQG community spin foams). The idea of spacetime foams dates back to wheeler. The idea is that at small scale the quantum fluctuations of the metric become very important and the plane spacetime disappears. In that "foam" could, at least it is cited so in the usual divulgation, happen unusual things, such as topological changes of spacetime. These could result in the virtual formation, and annihilation, of things such as black holes, and also, wormholes (a bit more on these later).

This is the naive view. Stringy theorists claim that S-duality changes drastically this scenario. The reason is as follows, when you get some string theory in weak coupling you can prove that it´s correspondent in strong coupling is another string theory (or maybe M-theory or F-theory in certain circumstances). For example, the dual object of a fundamental string is a D1-brane. Also there is duality between different branes (including duality between D-branes and NS5 branes). How this duality prevent the spacetime foam? It is somewhat obvious once one thinks about it. The strong coupling theory corresponds to the limit where the spacetime foam would appear and instead of it one gets another string theory. I have not had time to think to much about these, but I see some possible subtleties. The first one is that the S-duality is obtained by some heuristic arguments in perturbative theory which can be extended to non perturbative one with the help of supersymmetry. That raises the question of how much these dualities are related to supersymmetry or to D-branes. These could seem irrelevant, but not necessarily. In open string theories (for example, the bosonic one) D-branes appear as diritlech conditions for the extrems of the open strings. You don´t need supersymmetry and RR charges and all that for the existences of that D-branes. I have no notice of the search of string dualities for the bosonic string, but naively one would spect that still the S-dual of an fundamental bosonic string would, again, be an D-1 brane, but I can´t say for sure.

In fact all these could seem totally uninteresting to someone. I still think that to clarify the exact role of supersymmetry in the dualities is basic, after all pure (non stringy) supersymmetric theories also have branes and maybe there is some kind of S-duality betwen point particles and 0-branes and these would mean that supergravities also forbid the spacetime foam. But by now my concern about spacetime foam will go into anther direction, the (noncritical) Liouville strings.

If the reader makes a google search (s)he will find that in closed string theory there are also D-Branes, introduced as boundary states in the conformal theory (see, for example, my previous post about these theme). The interesting, for this post, fact is that also noncritical (closed bosonic) strings can be shown to have D-branes. If the S-duality depends exclusively in the existence of D-branes that would mean that there is no reason for the existence of spacetime foam in Liouville strings, contrary to the claims of Nanopuollos at all. On the other side if the S-Duality depends on supersymmetry it is expected that (super)Liouville strings (the relevant ones if the theory must reflects the reality which has fermions) would also have S-duality and so no spacetime foam. The obstacle for these would be the existence of D-branes, but as far as I see the Liouvile strings would have the same R-R sector, and charges, and so some kind of D-branes, in the appropriate dimensions, so one would expect definitively no spacetime foam. Well, that is what I expect previous to a careful reading of that papers, why didn´t i did it?

Well, that leads me to the last topic of the post, the wormholes. I assume by now that all my readers have the intuitive idea of what a wormhole is. I´ll make a separate post in this topic anyway. This time I´ll just say a few things. On one side one of the ideas of Wheeler is that they could be formed in the space time foam. After all the continuous evolution of a metric can, in general, drive one for spaces with a certain topology to others with a different one.

But if one studies carefully general relativity one learns that there are some theorems stating that if causality must be respected everywhere there cannot be such transitions in topology. Also quantum tunnelling between different topologies could be ruled out under certain reasonable assumptions. Wheeler seemingly ended by trying to achieve something different. Instead of absolute changes in topology you could search for effective changes, i.e. that the transition point between topologies would be of subplanckian size, but not a point. Anyway, I know that string theory claims that it allows topology changes in space time. And also it describes wormholes. In fact (transversable) wormholes relies for it’s stability in some special matter, or, in the presence of a positive cosmological constant. The discovering of these constant in the actual universe has launched an interest in wormholes in the string community, specially in the Randall-sundrum sceneries. I begun to study some papers, but I got somewhat lost and searched for some guidance. I have found very interesting, and very useful, the book by M. Vissier: "lorentzian wormholes" (1996, springer verlag). It is focused in "relativistics" viewpoint on white holes and I am not sure if he says anything about string theory. Also it is previous (I gues) to the discovery of the accelerating universe. And it is somewhat old. But still so it is being one of the most interesting books I have readed for a time, it is very well writing and results easy to understand, at least if you have a good basic in GR. When I would have readed this book (or a relevant part of it) I will try to read the papers of nanopoulous and comment on them. Oh, yeah, I am also reading occasionally a classical (1993) paper in closed string field theory inthe BV antifield formalism. I have confident knowledge that it is, even today, a relevant paper and that you need to read it if you are interested instring field theory. But compared to the Witten open string field theory, and it´s results about tachyon condensation I find these paper terribly boring, yeah, I know that is of topic to these post but...I needed to say it!!! ;-)

Maths and physics

Untill now most of the posts inthese blog have been expository. It is time for an expeculative one. I´ll try to give a few musings about the role of math in modern physic.

Before anything else I must say that it is totally ridiculous that while theoretical physics goes into more abstract mathemathics graduate programs of the universities goes towards less level in maths in favour of informatic skills. I don´t meanthat the abbility to program would be not important, but I guess a physics student can learn to program anywhere else and t it totally unnecsary to cope an asignature for that purpose. If somene thinks that the informatic in a physics or mathemathic faculty is anything special all I can say is that I learened to program out of any faculty and when I needed tto pass an exam (in the math faculty) I got a 10 note althought I didn´t go to the classes of the asignature, didn´t study a single minut of it, althoguth when I did the exam I had not programmed a single code line in C (the actual language asked in the asignature) for more than a year. So I am certain that not studing a math asignature to study informatic is a totall mess.

Well, afther these break about informatic let´s go with the actual topic. When I was studiying physic at the faculty I had the good sense to look what people was doing inn research. And the most obvious thing that i noticed was that the level in math was very far beyond that was beeing teached in the faculty. That, together with the fact that I hate the lack of rigour of physics in some aspects lend me to ampliate my math studies. I, firstly, studied some of it by myself (begining by set topology), later I simultaneated studies in the physics and math faculties. In particular I got all the geometry and topology related assignatures. In fact I later beguined to study both geometry and topology to an upper level that the graduate asignatures. Specially I loved diferential and algebraic topology. I also studied somewhat about funtional analisis an related topics (that is, measure theory) but with not too much enthsuaism and only to be able to understand a book, the Galdindo and Pascual two volume exposure of quantum mechanics which uses the full aparatous of hilbert spaces as studied by mathematicians. However I didn´t studied another asignatures which I didn´t find necesary for physics so I didn´t end maths at that time. I prefered, instead, to study things such like nonlinear dynais, which had not place in the curriculum of my faculty. A separate chapter deserves group theory. It was inteh physics faculty curriculum and it was teached along the lines of books such as the hammermesch and similar ones. I just can say that fore someone used to the manifold theory I found so terribly poor the exposition that I had to studi by myself ll the manifold part of the theory, it was a pitty that as cause of it I didn´t get such as good familiarity with things like representations, young tableaux and similars and I needed to relearn it later.

The price to study all these math was that I didn´t get preciselly a brilliant expediente in physicst. I never have got disapointed with that fact because I have found a lot more usefull to learn modern math that having a good and deepd knowledge of such things as electronic, optics, nuclear physics and such that.

But said all these in favour of maths I must point also some negative. The spirit with math and physics at the end of the seventies and the beguining of the eigthies seemed to be that it was neccesary to recover the lost time and that physic could win a lot using the modern math. I totally agree that formulation of general gravity in the old fashioned way, i.e. the tensor calculus of Levy-Civita is a total mess. The language of difernetial manifolds makes a crucial diference at the level of understanding the physical ideas behind general relativity.

Somewhat diferent for me is the formulation of gauge theories as conections in principal fibre bundles. Ok, it is a good tool for actual calculatons of monopole or instanton solutions, but personally I don´t see that it gives too much physical insight, if any, which you couldn´t get in the traditional formulations common to phyisicans. I have not gone too much into the moduli space stuff, beyond that teached in the string theory books, so I can´t say anything about it, butwith that exception I personally don´t find find as fastanstic these formulations as many people seems to think.

Well, in the previous two parrafes I have treated mainly the reformulation of stablished theories in a new language. But that is not all what physics could expect to gain from maths. Afther all the very born of physic such as we know it is intimately related to the born of infinitesimal calculus. The Newton laws simply couldn´t have existed if calculus wouldn´t have beeen created first. A few centuries later there was another physical theory whcih required of a, by the epoch, new area of maths. I speak, of course, of general relativity. Albert Einstein didn´t like when Minkowsky reformulated his special theory of relativity in geometrical terms. But without that reformulation it would have been probably imposible to have reached the formulation of general relativity in terms of tensor calculus, i.e. diferential geometry. That was the second case in history where a radically new branch of math played an important role in physics. But for around tw centuries physic could work with the stairght develoments of the math which had raised it to the existence, calculus. The third case where a new branch of math played an important role in physics was the matrice formulation of quantum mechanics.But said these the truth is that the schröedinger formulations in terms of wavefuntions and diferential equations was mcuh more important. It could be said that hilbert spaces, which for many old fashioned physics is not much more that a combination of linear álgebra and sturn liouville theory, play a crucial role in quantum physics. Well, sure, but still most introductory books in quantum mechanics don´t actually explain what a hilbert space is (for a mathematician taste, I mean).

The next big even where math was crucial for physical developments came from the hand of Murray Gellman and his "eight fold way". The aplication of group theory to make sense of the hadronic zoo had a crucial practical impact. I am no sure of how important group theory was before that. Now most people like to relate the Lorentz group to quantum field theory in an absolutely crucial way. But I guess that in fact It didn´t play a very important role and that the diferential equationsprocedure was mos relevant in the development. I.e. People had the klein gordon equation, and later the Dirac equation, and separatelly the Maxwell equations for electromagnetsim. The fact that they were related to spin 0, 1/2 and 1 representations of the Lorentz group was probably something very secondary. It was not until the introduction of grout theory in the flavour stuff of haronic physicis, and later in yang mills theories that group theory was realised as something important and usefull, but my particular viewpoint is that is importance is somehwat exagerated. In particular I think that grout theory allows some quick calculations that let people to play not as much atention to some aspects of the theories and probably something is lost in the process.

With the rise of string theory modern maths became crucial. At the begining math made a diference. Most physics didn´t know modern math and simply couldn´t follow the results, less to say to participate in the developent. But that times went and now in greater or less extend everybody is familaar with modern maths. These means that wht decides if some can make important contributions to string theory (or other aproachs to quantum gravity) depend more in the usual physical intuition and less in familiarity with abstract math. Althougth I must say that I am very skeptic about how appropiately some of the string theorist have learned modern math. Undoubtly Witten did it (his field medaill proves it) but not everybody is Witten.

I have invoked the name of Witten and I must say some more things about him. Altought some of his works are of aundoubtly physical utility many of them are mostly mathematical. For example topoligcal field theories (which I learened beofre string theory, remember, my faouvourite branch of maths was topology ;-) ) are mainly an aplication of the path integral to a topological problem. Althought TFT are a beatifull theory it´s physical utility somewhat dissapointed to me. But TFT´s are a somewhat special topci, what about the rest of physics? I am begining to belive that people still are in the initial beief that by simply appliying new branchs of math they would automathically get new physics. And it is not working, as somewhat would expect. In these point I clearly disagree with Lubos Motl. Le´ts go with an example. algebraic gemoetry. People didn´t learn algebraic gemetry and sudenlly decided to searchwhere to use it. It worked somewhat in the reveres direction. There was a proble, to make proper sense of compactifications in orbifold gometries and reomve some kind of singularities. And them algebraic geometry, and blowing up of singularities came to the rescue. B.T.W. if am skeptic of how properly physicans have learend some branchs of math, topolgy, difeential geometry, my doubts increase when we go to algebraic geometry (by algebraic geometry I understand it inhis full formalism, varieties in arbitrary fields, scheme theory, and not only the special case of complex diferential geometry, or it´s still most reduced subjecto of Rienman surfaces) . By now my understanding of it comes from what is teached in string theory books and some clarifications that a friend of me, who is doing a thesis in number theory (which requires hughs amounts of algebraic geometry) did to me of some aspects.

Anyway, the thing is that physics have almost exausted all the branchs of mathematics (they are using even some absturse areas such as p-adic numbers in things such as p-adic-or adelic-strings or topological geometrodynamics). I don´t think that looking towards the few remote areas which, maybe, still have not been exploited will make any diference (particularly I seriously doubt that category theory would be something which will give anything relevant to physics). Perhaps the only exception would be some areas of math which are somehwat beyond the usual scope of theoretical physicians, nonlinear systems and complexity (in a broad sense which covers things such as markov/stocasthic process, grahp theory, etc) could become relevant. These rise agian the role of group theory. Group theory is importan when symmetry is the key ingrediente. But in nonlinear sciences symmetry is not such important. Maybe playing more atention to nonlinear sciences could force to an small change of paradigm to theorethical phyisics or maybe not.


A place where certianly complexity should play a role (I am aware that some papers ahve already gone trought these line) is in the landscape problem. I totally agree with the skeptics about the utilitie of the anthropic principle. If there is no way to remove the landscape the apropiate tools to investigate it whould be complexity theories and not any kind of anthropic principle. I have had the luck to teach math to people working in biology and for sure all their lines of reasoning are much more addequat to trate all the landscape questions that that stupid anthropic principle. But I hope that someone would find, and if possilble soon, a diferent solution to the cosmological constant problem that the landscape idea.

But with that possible exception I gues that it is time for physicist triying to actually do physics and not relay on looking into mathemathcians to search for their new "revolution". Afther all there was a short epoch where the game worked in the opposite direction. I am talking about Dirac and it´s extensevely used "Dirac´s funtion" which when formalised by the mathematicians became the distirbution theory and about the Feynman path integrals whose proper formalization suposed a lot of hard work in measure theory. Physicians could use these math in the non formal treatement which mathemathicians developed later with a lot of success. In fact many still do it and don´t care at all about the more sophisticated versions. B.T.W. I mentined before that the fibre theory formulation of gauge theories wasn´t, in my opinion, too relevant for physics. But it has been very usefull for mathemathicians (there are a lot of docotrants doing his thesis about those topics). Also, seemengly, applies with some aspects of stiring theory. But I am not sure that these cases are the same that dirac delta funtion or path integrals. Another important aspect is whether most mathemathicians could understand relatively well classical physics and evenquantum mechanics (and certainly general relativty) but I seriously doubt they understand properly gauge theories, the QFT aspects of it, or string theory so the relation, or relevance of the interplay betwen these theories and maths, from the mathemathicals viewpoint is more obscure.


I have not been as organized in the exposition of the ideas that I wanted to express as I would have liked, but hope it sitll there is some coherence in the post.

The case for D-Branes in closed strings: Boundary States

In a recent post about the "brane forest" I said that although in type II strings, which are closed strings, there are d-branes, and play a very important role, I had not seen an explicit construction of them.

After all in open string theory there is an easy way to see the appearance of d-branes, they are hyperplanes at which the extremes of open string can end. But, clearly, the same image badly could work in closed string theory because closed strings can leave a d-brane (these is the stringy inspiration for the wrapped brane sceneries ala Randall-Sundrum). It has not been totally trivial to find a minimally satisfactory answer (for my taste) of the question. Before going to it I´ll review some heuristic arguments.

The most common one is based in the analysis of d-branes interactions. These can be understood in different ways. One is by the mediation of open strings joining the two d-branes. Considering a one-loop diagram of these strings is equivalent to consider a tree level diagram of a closed string going between these two d-branes. Ultimately these is related to the fact that poles of open strings diagrams of open strings would correspond to closed strings and that, maybe, open strings alone would not be an self-consistent theory and would need a closed string sector.

Well, anyway, these is as far as many articles and books who review d-brane theory go in these picture. They present some other arguments about the existence of d-branes in closed strings, but before commenting on some of them It is time to present the way in which these picture can be achieved in a math formalism which goes beyond wordy arguments. The trick is the use of a an artifact of conformal field theories known as "boundary states" (for the sake of truth I must say that the book on d-branes of Clifford Jonshon actually mentions it, but is far from presenting it in any clear way). I we go to the original literature we can see that the original paper of Polchinsky on the subject http://arxiv.org/pdf/hep-th/9510017 uses these formalism. For people not familiar with conformal field theories, and particularly with the boundary state formalism I leave a few links:

http://wildcard.ph.utexas.edu/~shaji/papers/misc2.pdf (easy to understand intro to boundary states, it requires previous knowledge of CFT)

http://arxiv.org/pdf/hep-th/0011109 (General introduction to CFT´s, including boundary states, it is more formal/rigorous than the previous. As an advise for casual reader to comment that what in the first paper is called "method of images" i these is called "Stocky conditions").

IF someone doesn’t want to read these papers he can get a "quick version" in the original paper of Polchinsky or, for example in http://arxiv.org/PS_cache/hep-th/pdf/9707/9707068v1.pdf (additional papers could be http://arxiv.org/PS_cache/hep-th/pdf/9510/9510161v1.pdf or http://arxiv.org/PS_cache/hep-th/pdf/9510/9510135v2.pdf).

The argument is as follows, you can consider a diagram of closed strings and impose that the "out state" is not given by a closed string but an BRST invariant operator |B> that inserts
a boundary on the world-sheet and enforces on it the appropriate boundary conditions.

I’ll not go further in the details, and refer the reader to the literature cited before, specially these paper. Just to mention that boundary states utility in string theory is not only to introduce D-branes but are useful for many other purposes (for example in the CFT treatment compactifications). In CFT terms a boundary states has a role very related to it’s name, it imposes conditions in the boundary of the world-sheet. Just for completitude I´ll mention that boundary state formalism is a very "natural" way to introduce D-branes in the string field theory version of strings theories. I hope to write some entries in SFT so I´ll give them the details.

Well, these is the most rigorous way to introduce d-branes in closed strings. But in most of the literature it is not made, why? Probably because it is supposed that CFT is reserved to the "advanced" reader, or maybe because supposedly the other ways are in some why "more physical". For example in CFT you can get a more rigorous way to achieve effective actions for certain approximation of situations of string theories. Aa very related example, the Born-Infield action of the d-brane) but the picture of the fields associated to the extremes of the open string dictating the physic of the branes is more appealing.

Well, these are the more rigorous way, but I had said that there were more considerations that support the existence of d-branes in closed string theories. In the original paper of Polchinsky, after the boundary state formalism treatment, he offers an interesting argument by referring to the Hilbert space which I reproduce here:

Periodically identify some of the dimensions in the type II string:

X^u= X^u + 2πR n=p + 1,...,9 (for a Dp-brane)

Now make the spacetime into an orbifold by further imposing

X^u= -X^u

To be precise, combine this with a world-sheet parity transformation to make an
orientifold. This is not a consistent string theory. The orientifold points are sources for the RR
elds (by the analog of the above arguments for D-branes, but
with the boundary replaced by a crosscap *), but in the compact space these
elds have
nowhere to go. One can screen this charge and obtain a consistent compacti
cation
with exactly 16 D-branes oriented. Now take R->0. The result is the
type I string

A few more comment to conclude. An interesting aspect of the boundary state formalism is that you can get a somewhat dual vision of the paper of the d-branes. Instead of considering interactions between them mediated by strings you can think on them as solitonic solutions of string theory and doing perturbative string theory around that solitonic background. For a review (previous to the introduction of d-branes) in string theory the reader could read these article, String solitons.

I, personally, haven’t still readed it, i have a lot of lectures to do, as well as many things to think about. But one thing, the last one of these post, that I wanted to comment is the following. In the viewpoint of a d-brane as an hyperplane where an open string can move semengly is obvious that a d-brane is an infinitely extended object. The full secenarie of Wrapped universes of Randall-Sundrum (or the ekitropic scenario) support these viewpoint. But as is well known d-branes are not supposed to be rigid objects because in gravity there are not such objects, and in fact, guided by the p-brane scenery of supergravity, and also by the theory of "fundamental" branes you can think of a d-branes as a generalization of an string. From these viewpoint it is not clear at all that the brane as an infinitely extended object would be the right one. In fact the 2 and f5 branes of M theroy (M-branes) are supposed to be fundamental objects (despite the fact that it is very difficult to make sense of a theory which gives an interaction picture for them) which in some limits become ordinary fundamental strings. But ok, M-branes are not d-banes, is there a picture to think about d-branees like "small" objects?.

Well, in ordinary QFT there are solitonic solutions to Yang mil theories. They can be interpreted as monopoles. And they are particles with a finite size and all that, certainly not infinitely extended objects. If-d-branes are solitones for an string it would be natural to think of them as small objects of a certain size and not, at least without further reasons to think so) infinitely extended objects. Even if you think in the hyperplane picture you have that they are hyperplanes for a single string, but, why all the strings would be constrained to the same hyperplane?. If you go to the study of d-bane interactions you see that parallel static d-branes don´t interact, but that when they form some angle they do, and, for example, break some supersymmetries. Also moving parallel d-branes interact with a potential proportional to they speed (these configurations of d-branes can, for example, be used to probe distances smaller than the string size). Books, and reviews, explain (or actually prove) these results, but they don´t mention too much about what to think about their implications. I guess that it we must think about it it could be that d-branes related to the many individual strings in the universe, interact to recombine into a single "big" d-brane, or maybe a few parallel ones, but it is conceivable that "small" d-branes with arbitrary size exist. In the literature of string theory you frequently read about d-branes wrapped around torus (or K3 spaces) when considering black hole entropy counting (i hope to write an entry abou these sometime in the future) but it is not so usual to read about actually "shaping" d-branes. Al I have readed so far is in the divulgative book of Leonard Suskind "the cosmic landscape" stating that d-branes in spherical configurations are instable. In a more formal way the d-branes book of Clifford Jonshon in the last part of the chapter in "d-brane geometry I" consider non hyperplanes configurations of d-branes, related to ALE spaces (asintotically locally eculidean), but it a very indirect treatment and it is not easy to follow the details.

Well, as the reader can easilly deduce I am not an expert in string theory (see, for example, the blogs of Lubos Motl or Jackes Distler linked in these blog if you want ton consult the expers) but I hope that maybe reading to someone who has not all the answers can be also interesting because, maybe, the reader could have arrived himself to the same doubts I have, and maybe, I could have answered some of them partially. Also I think it is interesting to discuss somewhat "old" topics and not only the last papers in arxiv (altlthought I don´t discard to discuss some, of course). In any case, fortunately, these last times I have considerable amounts of time to dedicate to the study of strings (and in less extend other approaches to QG) so I hope I will become a more reliable source of information in the near future ;-).

About the "roads" to quantum gravity

As most peoople who is nowadays interested in quantum theories of gravity knows, there is in the internet something called "string wars". These "wars" consist on a dialectic batlle betwen some string theorists, with Lubos Motl as the main contendient vs some people who claim that string theory is dead (mainly Peter Woit) because of it´s lack of experimentally falsifiable predictions and other people (mainly Lee Smollin) who defends that even if string theory is a legitimate theory there are another aproachs to the problem of the quantization which deserve more funds, mainly LQG.

In these blog I have had entries about strings, LQG, NCG and (the last one) conformal gravity, which in greater or lesser extent claim to have something to say about quantum gravity and links to some pages about these topics. Also I have linked the Mitta Pitkannen blog (http://matpitka.blogspot.com/) which is devoted to "topological geometro dynamics" about which I havent made any entry simply because I don´t know enought of it to do so, but which is (or at last I tnink it is) another proponent as quantum gravity.

These doesn´t exaust the list of "roads" (an Smollin terminology) to quantum gravity. For example a very famous and well considered physics, Roger Penrose, thinks that maybe when he would be able to develop it enought its twistor theory could be a candidate as a quantum gravity (nowadays it is mainly a math tool apropiate for some tasks, but very awfull for many others). If someone is interested the other famous English physician, Stepehn Hawkings, defends his own aproach,euclidean quantum gravity. I have knowledge about a few others but I am not interested here in making an exaustive list.

One could answer, and it is a very reasonable question, why to bother with so many theories? Afther all string theorists insist in that althoguth they have not experimental avals they have made many self-consitency tests and they are sures that string theory is the only "game in town". Well, of course these sounds like a "no-go theorem" and as any other no-go theorem it is as solid as the weakest of its asumptions. I am not going to go into the details now but the main point in these claim is that string theory represents an scenary where quantum theorie, just as it is nowadays known, is preserved. On the contrary LQG people would defend the gravity expert viewpoint.

Other theories, like NCG (non conmutative geometry) could have a place as "effective theories" in some range (these is the way string theoriests seem to think about it) or as a separate theory (the Alain Connes viewpoint).

I have not intention in these blog to get any position in these wars and neither to give any support to personal attacks in the line of "crankpots" pursuit (at least not for academically acredited people, "Subiiris" or "Paulinos" will have zero tolerance) . Thse doesn´t mean that I don´t have my own opinions, nor that I consider equally likelly all these aproachs, and of course I will not dedicate the same time to learn all of them. But I think it could be interesting to present, at least in overview from tiem to time, some of the less known theories. I guess that an inteligent and prepared enought reader can make his own idea of how viables these theories are.

Conformal gravity, a new theory of quantum gravity?

I have just seen in physics forums the following paper:

Conformal Gravity Challenges String Theory

I had no previous knoledge of these theory and now I have no time to search in google references for it so I just expose it without any claim about how good or flawed it could be.

The author, Philip D. Manhein, semmengly a refuted cosmologist, reviews the genesis of general relativity and isolates two parts, kinematics, background independence, and dynamics, Einstein equations. The point is to search an alternative dynamics. The ultimate reason for Einstein equations is a fenomenológical law, the Newton potential V=1/r. These can be shown to be a solution to the Poisson equation. But if we want to allow small variations compatible with actual observations we could go to V=b/r + c.r with a very small c. A solution of these kind can be shown to be compatible with a fourth order derivative Poisson equation:



In background independent terms we can make a theroy based in the Weyl (conformal) tensor C^uvnk form which we can derivate equations of motion of the form:

4αW^uv=T^uv.

These equations have Schwarschild type solutions and some other aspects coincident with Einstein theory. Even thought the most important concern, of course, are de diferences with Einstein which are basically 3:

1)At galactic scales the mass distribution deduced from the apropiate equations fits the observed distribution without requiring dark matter (here I would point that recently seemengly there have been indirect observations of dark matter so maybe these could be a problem afther all).


2) At cosmologic scales an equation equivalent to the Friedman-Robertson-Walker (with an apropiate energy moment tensor) can be formulated and we obtain a solution whic describes an aceleratedly expanding universe withouth a cosmological constant (which is forbiden in that theory because of conformal invariance).

3) It is a power counting renormalizable theory. That means that if we construct a perturbative quantum gravity from it it could be renormalizable, i.e. fully consistent. And it would be a 4 dimensional theory, no need for extra dimensions.

Until now all seems very correct.But beeing a relativelly easy theory (as compared to string theory for example) one could think that there is some sublety involved and in fact that was the case. If one calculates the propagator for the gravity sector one finds a term which from knowledge of quantization of gauge theories seems to be associated to a ghost state which without some apropiate way to handle it would remove the unitarity of the theory. Well, the author in these paper, and these is the important development here, claims to have resolved that problem, which seemed to be a generic problem for theories with fourth order derivatives.

I´ll try to read the article more carefullly, and of course also wait for posible reactions in the physic comunity. All it sounds very interesting but suposedly one would be carefull with "fundamental" theories developed by a cosmologist. Well, the paper was brief so in the worst of the cases it didn´t mean an excesive mess of time.

P.S. String theorists claim that there are two factors which seem to indicate that stringn theory must be the "only game in town". One of them is that a quantum gravity would be a fixed point in the renormalization flow defined by the beta funtions of the theory if it going to advoid the need of an innite number of parameters. You can read a carefull exposition of the argument in Jackes Distler blog, concretelly here. Well, conformal theories have ultraviolet fixed points and these is a conformal theory so it would fit the requirement (in that link Distler claims that string theory escapes the problem by a diferent method, even though string theory seen as a conformal theory on the world sheet fits the requirement of ultraviolete fixed point, I am not sure if I am mising some point with these two appearences of UV fixed point from two slightly diferents viewpoints)

I am also aware that something called Poisson deformations (or something similar) seems to indicate some uniquiniess of string theory. I dón´t know how the result is obtained and the secenaries it covers so I can´t judge it´s relevance for the present theory.

Update Afther a bit of search I have found that there articles which have the details of the calculations:

http://arxiv.org/abs/astro-ph/0505266 (classical part)

http://arxiv.org/abs/0706.0207 (quantum part)

You can read another opinion about the article here:

http://www.scienceagogo.com/forum/ubbthreads.php?ubb=showflat&Number=21054&page=2

The theory was presented in the recent Pascos 2007 conferences and seemengly it had a good aceptation and now people is studying the details of the math in search of possible faliures. If not found seems it definitivelly looks like a promising theory.

String field theory para dummies

Estos días he estado mirando cómo anuncié, entre otras cosas string field theory. Voy a intentar dar una idea muy superficial de en que consiste y ya expondré los detalles en algún otro momento.

En un post anterior expuse con un cierto detalle la parte clásica de la cuerda bosónica, escribiendo su lagrangiano y explicando algunas de sus características (invarianzas y demás). Cómo expliqué allí ese lagrangiano era poco más que una generalización del lagrangiano de una partícula relativista a una cuerda relativista. Si cuantizo ese lagrangiano (o su análogo supersimétrico) de alguna de alguna manera se obtiene la teoría de cuerdas convencional.

Una forma de ver la cuantización es la sustitución de la posición x y el momento p por unos operadores correspondientes que actúan sobre un espacio de Hilbert y que obedecen una serie de condiciones de conmutacion [Xi,Xj]=0, [Pi,Pj]=0, [Xi,Pi]= ihδij.

Bien, para la curda bosónica se pueden tomar como X´s as coordenadas del centro de la cuerda y cómo P´s los momentos de la cuerda e implementar esas relaciones de conmutación. Esto no siempre es transparente porque normalmente se ha hecho una descomposición en modos de Fourier de los modos de vibración de la cuerda y se proceden a imponer relaciones de conmutacción para los coeficientes de Fourier del estilo de las del oscilador armónicos, es decir, pasan a ser operadores de creación-aniquilación. Pero puede verse fácilmente que ambas dos cosas son la misma, usar coordenadas del centro de la cuerda o modos de Fourier son lo mismo.

Una teoría cuántica dónde se ha producido una cuantización de las coordenadas y el momento es una teoría de primera cuantización. La ecuación de Schröedinger de la cuántica no relativista es una teoría de ese tipo. Las ecuaciones de klein Gordon y de Dirac para partículas relativistas son de ese estilo también. La característica más destacada de esas teorías es que son teoría de un número fijo de partículas. Las interacciones entre esas partículas se introducen mediante potenciales dependientes de la posición (normalmente) de la partícula, cómo puede ser el potencial de un oscilador armónico (el de la ley de hook) o el de la ley de Coulomb.

Sin embargo hay casos en que el número de partículas no esta determinado. En cuántica no relativista esto ocurre para sistemas de materia condensada, para mecánica relativista el caso es mas serio pues la famosa relación de Einstein E=mc² implica que a energías lo bastante altas pueden crearse partículas a partir de la energía. Digamos que las ecuaciones de Klein-Gordon y Dirac sirven para partículas que se mueven lo bastante rápido para que haya correcciones relativistas al potencial Coulombiano y cosas así, pero no lo bastante rápido para que la energía de las partículas puedan crear unas nuevas. Por ese motivo hay que ir más allá de esas ecuaciones. Lo que se hace es una segunda cuantización. Recordemos que una partícula, relativista o no, se describe en cuántica por una función de onda Ψ. Antes comenté que cuantizar era sustituir posición y momento por operadores. Todo el mundo suele tener en mente que el momento es P=m.v. Bien esto es así para el caso no relativista. Hay una expresión para el relativista que no pondré aquí pues lo importante no es su expresión exacta sino explicar que es el momento. El momento es técnicamente una cantidad que se obtiene del lagrangiano que representa elsistema físico por derivación respecto a la velocidad (derivada de la posición respecto al tiempo).

1.

Las ecuaciones de Klein-Gordon o de Dirac pueden obtenerse a partir de un Lagrangiano. Es un lagrangiano que dependerá de la función de onda, que puede considerarse en cierto modo cóm un campo. De hecho si consideramos la interacción electromagnética de manera sofisiticada en lugar de un lagrangiano tipo ley de Coulomb y similares tendremos un lagrangiano en el que aparecerá el campo electromagnético. Tenemos pués una teoria de campos. El campo para el boson (klein-gordon) o fermión (Dirac) se había obtenido como una primera cuantizacion de una partícula clásica, pero podemos olvidarnos de ello y pensar que es un campo. El campo electromagnético es, en principio, un campo clásico, pero lo importante es que ambos pueden considerarse cóm ocampos. Y las ecuaciones de esos campos surgen de un lagrangiano, un lagrangiano de campos. Pués bien, una sencilla generalizacioin de la ecuación 1 nos da la definición del momento para ea teoria. Claro, no es, en principio, algo que tenga la forma m.v pués lo que nos van a a aparecer serán expresiones que dependan de los campos. Pero aprte de eso formalmente se puede seguir el mismo procedimiento de sustituir coordendas (los propios campos) y sus momentos asociados por operadores. Los detalles obviamente no son triviales, estos operadores actuan en un espacio de Fock y no en uno de Hilbert y etc, etc. Perolo importante es que formalmente puede hacerse. Lo importante es que esta teoria va a describir sistemas dónde el número de partículas no esta determinado. Además ahora lo que nos a a interesar es obtener lo que se conocen como secciones eficaces, qeu se obtienea partir de elmentos de matriz S. Sin entrar en detalles esto viene a indicarnos la probabilidad de que al chocar un cierto número de partículas incidentes aparezcanan cierto número de particulas salientes (que pueden ser o no las mismas que inciden) y en que ángulos se desvian.

Bien, esto para partículas. Las cuerdas son un mundo aparte. Recordemos que es una funcion de onda, Es una funcion Φ(x,y,z,t) cuyo cuadrado nos da la probabilidad de hallar la partícula en las coordenadas x,y,z en el tiempo t. Uno podriá preguntarse cuál es el análogo para las teorias de cuerdas de esta funcion de onda. La verdad es que normalmente los libros de teoria de cuerdas no suelen comentar nada al respecto y no siempre deviene uno en plantearse esto. Al fin y al cabo quien estudia cuerdas ya conoce teoria cuántica de campos y en cuanto ha visto por ahí circulando los α´s y sus conjugadosque son operadores de creación-aniquilación y le han introducido un concepto de vacio y le explian que una cuerda tiene varios modos de vibración y que cada modo de vibración es una particula diferente tiende a pensar que esta en una teoria cuántica de campos con muchas partículas y se preocupa por cómo se calculan elementos de matriz S, y además esta por esos entonces asombrado con otras cosas, cómo las dimensiones adicionale, la aprición de taquiones en el espectro, o intentando ver si entiende de una manera razonable cómo ese mesmerismo de teoria de grupos conduce a la identificacionde modos de vibración con partículas y etc, etc.

Pero en el fondo, tenemos una cuerda vibrando que representa, según vibre, una partícula, o para ser exactos, tenemos un número determinado de cuerdas que reperesentan un número determinado de partículas. Pero, en principio, el sistema no representa un número arbitrario de cuerdas.

Así pués, sabiendo que tenemso una sola cuerda si tiene sentido plantearse si tiene sentido construir una funcion de onda para la cuerda cuyo cuadrado pueda interpretarse cómo una probabilidad cuántica. Tiene sentido, pero casi no hay nada hecho al respecto, sólo he visto algó de un tal Nathan Nikovits (qeu poste an physics phorums como demystifier) pero usa una interpretacion un tanto rara de la funcion de onda basada en unas ideas de Bohm ue desde luego no son las más ortodoxas.

Uno podria plantearse cómo es posible tal cosa. çla respuesta es muy sencilla. La matriz S y el cálculo de secciones eficaces, osea, la teoria de colisiones cuantica, tiene sentido también en primera cuatización. Y en esa teoria de colisiones la ecuacion de Schröedinger y la interpretacion en términos de funciones de onda no juega un papel relavante, lo importante es tener un Hamiltoniano (que contien la msima información que el Lagrangiano). Así pués digamos que la teoria de cuerdas "ordinaria" en cierto modo es hacer teoria de colisiones de cuerdas cuánticas en primera cuantización. Esto desede luego no es en absoluto transparente en el formalismo. De hehco normalmente las interacciones entre cuerdas no se introducen mediante un hamiltoninao o Lagrangiano sino que se hace una prescripción, la integral de Polyakov, basada en una analogia muy formal de la suma de caminos de Feyman y se sustituye la suma sobre trayectorias por suma sobre superfices. En realidad, y esto es algo que casi nunca se ve aunque viene en el megaclásico libro de Green-Schwarz-Witten, se pueden deducir los términos de la integral de Polyakov mediante un lagrangiano de interacción, aunqeu la verdad es uqe no he visto aún todos los detalles.

Bien, esto es la teoria de cuerdas en primera cuantización. Deberíamos plantearnos una segunda cuantización si queremos tener una teoria sensata, al menos eso se pensaba con total firmeza en los inicios de la teoria de cuerdas. Esa teoría de segunda cuantización, una "string field theory" debería permitir entre otras cosas hacer cálculos no perturbativos y dar una guía para un vacio que representara la elección de la compactificación (no confundir ocn el vacio sobre el que actuan los operadores α que representan los modos de vibracion de la cuerda).

Ahí si que hay hecho bastnate trabajo, ahí si se introduce una "función de onda de la cuerda". Eso sí, a ser segunda cuantización no se analiza su interpretación probabilísitca, sólo se ve com algo formal, un funcional, que permita escribr un lagrangiano.

Tras varios trabajos y diversas teorias en gauges particulares Witten creó una teoria para la cuerda abiertaen la que el lagrangiano tenía la forma de una teoria de Chern-Simmons donde el producto ordinario entre "funionales de cuerda", Ψ, se sustituía por un "prodcuto no conmutativo" * del estilo al de las geometrías no conmutativas.

2.


En realidad la historia es un poquito más complicada que todo eso. Ψ es un elemnto de un "álgebra graduda", Q es un operador de derivación que luego se puede ver qeu se corresponde con lo que se conoce cómo "operador BRST" y etc, etc, pero en cierto modo implementa esa idea. Por cierto, ese lagrangiano 2 tan elegante expresado en témrino de productos ordinarios (conmutativos) tiene una forma horrorosa.

La teoria de la cuerda bosonica abierta ha tenido un cierto éxito. Por ejemplo a finales de los noventa se consiguio ver que las D-Branas podian considerarse como soluciones solitonicas de las ecuaciones de movimiento (para llegar a ecuaciones de movimiento se hace un desarrollo en serie que lleav a un conjunto infinito de ecuaciones diferenciales acoplados) y eso permitio medio demostrar una conjetura de Sen, obtenida en toeria de cuerdas normal, sobre que cierto tipo de sistemas D-brana/Anti D-Brana se podina aniquilar y dar un taquión (taquion condensation) y eso podia en cierto modo analizar la inestabilidad de la teoria de cuerdas no bosonica y explicar que pinta ahi el taquion del espectro y cosas así.

Eso para la cuerda bosonica abierta la más fácil. Witten sugiere que puesto que dos cuerdas abiertas pueden interactuar y formar una cuerda cerrada la cuerda cerrada debería surgir dentro de una teoria de curdas abiertas cómo cierto tipo especial de estados. No sé muco dónde ha ido a parar esa idea, pero en todo caso sé que hay una teoria específica de cuerdas cerradas, o de hecho es posible que varias. De un lado tengo noticia de una en la que trabaja sobre tod o Zweibach y colaboradores. De otro lado sé que Kaku y colaboradores trabajan también en el tema y creo que son teorias diferentes. Sé que en la "versión Kaku" estan intentando ver que pintan ahí las D-Branas, pero es algo con lo que estoy aún empezando a familiarizarme.

Aparte esta la supercuerda, o cuerda con supersimetría. Creo que lo mejor que hay en ese campo es la teoria de Nathan Verkobits para la supercuerda abierta. Es un análogo relativamente directo a la teoria de Wittern, solventando los "pequeños detalles" que Witten auguraba. En realidad más que por un término tipo Chern Simons termina usando algo del estilo de las teorias del tipo Wess-zumino Witten. Pero ya meterme a intentar explicar minimamente eso se me hace imposible, entre otras cosas porque aún tengo unas cuantas dudas al respecto.

Cómo curiosida dmencionar que para la supercuerda existe otra formulación basada en el formalismo de Twistors y una correspondiente string field theory. Y es esa teoria la que ha estado impicada en los recientes papers que sugieren que la supersimetria par partículas puntuales puede ser renormalizable en contra de lo esperado. Igualmente aún me falta un tanto para entender bien esos aspectos.

A brief survival guide for the brane forest

First of all a quick clarification about the use of two diferents languages, english and spanish, in these blog. Initially I had the intention of using only english, but my participation of some spanish forums about physics derived in posts which I find could be interesting here (in a more complete form that the original ones in the forums). Also in spanish there is less material available about high end physics and not everybody in Spain has good enought level in english to read easily in thath language.

Well, said these I go with these post. When studiying string theory nowadays you find a lot of branes going around and also aparently diferent meaings for the same type of brane. I think that it could be interesting to have a fast guide where you can have a reference of what is everything. As far as I couldn´t find any I have decided to try to write it. As the subject is very extensive I will not goo too deepd in the math details.

Well, let´s beguin by the most basic one, the p-brane. I´ll give first the most broadly used acception of the term. In string theory you have the Nambu-Goto action (see previous post if you speak spanish) which is a generalization of the relativistic particle action. If you allow one-dimensional objects, why stop there and not do a theory for p-dimensinal objects? Mathematically is easy, you simply need a trivial generalization of the Nambu goto action. fo rexample a two dimensional p-brane would be a parametrized surface an son on. In general the action is S= T. V where T is the tension (energy density) of the brane and V is it´s volume.
In fact there are some subleties and you need a cosmological term (see, for example, the Becker-Becker-Green book).

Well, we have a classical action for the p-brane, but if you try to make a quantum theory of it you run into deep problems. Even if you save them for the noninteracting theory you still would have the "small" problem of introducin interactions betwen branes, it is belived that such thing is not possible.

Ok, we have defined a p-brane. But if you go into the literature you find diferents definitions. For exaple the Michio Kakus book "string theory and M-theory" introduces the same terminology with a diferent meaning (also Bachas in his lectures uses the same terminology). I will explain it and i´ll go from there to another famouse branes, the D-branes.

In the midle of the ninties there was a problem with the type II superstring theories. In their spectrum there were antisymmetric fileds coming from the R-R (Ramond-Ramond) part of the spectrum wich are somewhat analogous to gauge fields. It was known that these fields would be charged and that meaned that it was necesary a source for them. The problem was that an string couldn´t be that source. The reason of it is that if you see that fields like a diferential form of diferential geometry is trivial to understand that it must an extended object of diferent dimension than an string. Concretely an C^p+1 field would couple to an extended object of p dimensions. Well, one could that the p-branes I defined previously could do the job. But as I said there were some problems with that branes so in that days people thought that the sources could be black p-branes, which are higher dimensinal analogous of black holes (more on these later).

Well, in fact, as the atent reader could have deduced, these p-branes couldn´t be exactlly the same ones that I introduced firs. One reason for these is that thes branes are charged and in the prevous ones there was no charge. You can introduce charge into these branes adding to them a term similar to the electromagnetic tensor. These takes as into another aspect, in electromagnetism you have electric charges and for the hodege dual of the electromagnetic field you have magnectic charges, that means that you can have electric and magnetic branes. Another thing to consder is that in an extended object the charge is spared. The total charge of the brane can be calculted using the generalized gauss law for a closed surface sourrounding the p-brane. There are many detaills about these, but I guess they are inapropiate for the purpose of these post.

I am going now to introudce the most famous of all branes, the D-branes. They can be introduced from the previous viewpoint and it can be shown that a p-brane can be made piling together d-branes, but I will follow a diferent way.

In open bosonic string theory you can impose Neuman conditions in the end of the string.But it also is possible to impose Dirittlech ones in some of the coordinates. That means that the string can move freelly in the Neuman coordinates but not in the Diritlech ones. If you have Diritlech conditions in p coordinates you have an string that only can move in an p-hyperplane. That is an extended object of p dimensions, and because it is related to Diritlech conditions it is named a Dp-brane where the p indicates the dimension.

Sometimes bosonic p-branes are introduced from T-duality. When performing T-duality in closed strings you get the winding number of an string around the wraped dimension. If you make the analogous and you take R->0 limit you find that the T-dualized open string is efectively constrained to move in one less dimension that the original one. T-duality interchanges Neumman for Diritlech conditions.



These is the very basic idea of p-branes, but I will explain a bit more about them in order to connect with another aspectos of it. In open strig theroy you can associate representations of field theories to ther extrems throguht chan-paton factors. If you do that some new aspectos for D-branes appear. On one hand the brane where the string end becomes charged under the gauge field which the string carries. Another aspect is that it allows that an open string could have their extrems in two diferent D-branes, the way to prove these requires Wilson lines and I´ll not even try to explain it.

Now that we have charged branes we can make a connection with the previous picture of p-branes as sources of antisymmetric RR dields. The idea is easy, you simply can pile together charged D-branes to fit the charge required for the p-brane. There are a few subleties wih these. For example nothing in the p-brane picture requires them beeing hyperplanes but D-branes appeared as such. The solution to these dilema goes back to a characteristic that I had not considered yet. Superstring theory is suposed to be a theory of gravity and in gravity theories you cant have stricitly rigid objects, that means that somehow D-branes mus become dynamical objects. You can go trought these considerations an obtain an efective lagrangian for the perturbations of the d-branes, the Dirac-Bron-infield one. An interesting aspect of it is that the dynamic of the brane is gobernated by the strings ending on it, but I will not go further with these.

Now I´ll itrouduce another viewpoint for D-branes. Superstring theories can be aproximated by effective actions. An efective action for a theory is a classical lagrangian which takes into acoount quantum effects (are tree level)of the original one. For superstring theories these can be done in many ways, for example finding a point particle theory whose amplitudes reproduce the string amplitudes (calculated throught the Polyakov prescription).

The important thing here is that the efeective lagrangian for superstring theories are supergravity theories. You can search solutions to the supergravity theories with some characteristics. I´ll motivate how d-branes appear in these picture. These will lead me to black holes. The most basic one is an Schwarschild one. A generalization of these is to consider a charged (under some gauge field) black hole, these is the Reissner-Nordtrom black hole. You can also look for black hole solutions in supergravity theories. If you search generalizations of these solution in superior dimensions you have what is called a black p-brane.

One interesting aspect of these black p-branes is related to the number of supersymmetric charges. I will not gohere deep into supersymmetry aspects and i will only give a very baci notions. Supersymmetry relates fermions with it´s supersymmetric partners. In the most basic theories you only have a symmetry, but you can have more, if you have one supersymmetry you have an N=1 supersymmetry theorie snd son on. The infnitesimal generators of the symmetri transformations are related throught commutation relations to the generators of the Lorentz group. That imposes an upper boudn of the number of symmetries that you can have in a ginven dimension, for example in four dimmensionsn you can have a maximun of 4 supersymmetries. In fact supersymmetry is broken in the real universe and there are strong reasons to belive that there is only one broken supersymmetry at low energies.

As I said I will not go far into supersymmetry, but I nedded a few basic notions to be able to introduce an important notion. It can be swhown that the black p-brane solutions have half of the supersymmetry of the theory to whcin belong (it is a common thing that solutions of a theory have less symmetry that the actual theory). In general one could be interested in searching for solutions with half the supersymmetry. That solutions are known as BPS states. The BPS states of the supersymmetric theories associated to a superstring theorie can be whown to have the same properties of the p-branes (d-branes) associated to the RR gauge fields I talked before.These shows the aspect of D-branes as BPS states.

Some puntualizations must be made here. I have introduced a pictorial idea of d-branes for the open bosonic string while all the other viewpoint implied closed superstring theories. These means that the D-branes of superstring theories are a generalization of the ones related to the open string theory. An explicit lagrangian for a super p-brane can be made generalizing the p-brane one to superspace. Superpspace is made adding to usual coordinates "supercoordnates", i.e, grassman type coordinates. For p=1 the p-brane is the Green-Scwhartz action of the superstring which is manifestly target space supersymmetric (not like the RamondNeveu-Schawartz one) but it is very ugly to be used in anypractical calculation. A most obscure point is that in the supersymmetric case we had closed strings. If we must keep the analogie these wouuld imply the existence of an open string sector in Type II theories. I hae seem in some papers stating that these is possible but I have not seen an explicit construction. Recently I have seen that people in string field theorie is triying to annalize these from a diferent viewpoint, but I still don´t know too mucho about these.

Untill now we have seen generic p-branes, black pbranes and D-branes. It is time to expose one common propertie of branes. One could think that is thses objects exist they could be important in string perturbation theorie and thay one would need to care about event in whcih an incoming string goes into outgoing branes an so on. In fact these doesn´t happen. The reason is that the mass (or tension, both are related) of the d-branes goes as 1/g where g is the string coupling. These means that for small coupling, the range in chich perturbation theory works, their mass becomes infinite and don´t appear. In the non-perturbative range both branes and strings have similar importance (In fact there are one dimensional D-branes, known as D-strings).

I have not gone into the properties and utility of D-branes. A quick summary is that you can wrap D-branes so that they get geometries very fr from the hyperplane. Thhey are tranformed trought dualities into other branes. Strings betwen diferent branes have a mass which depends on the separation betwen them. D-Branes parallel don´t interact betwen them. You can use apropiates combinations of wrapped D-branes and strings to construct Reissner-Nordstrom black holes and you can reproduce the Haking entropy of them. But counting microscopic states of excitations of strings betwen branes you can have a microscopic description of the black hole. The calculation of these entropy leads to the former implementation of the ADS/CFT concjeture and many more things. But a correct explanation of these subejects imply an understanding of modern string theorie, and that is something that you couldn´t expect from a simgle blog post ;-).

The ones that I have presented till now are by far the most common used branes but there are more, I´ll trate briefly some others.

I´ll begin by the NS-branes. All oriented strings have a common sector consisting of a graviton, a dilaton and a massles antisymmetric tensor field usually dennoted as B_mn. For similr reasons that ofr the RR fileds you can worry about the source of the charge for these field. For the "electric" charge the source can be shown to be the same string, but for the "magnetic" charge these must be an extended object. It´s dimension can be whown to be 5 and it is known as the NS5-brane. For the shake of completity I will mention that analogously as how you can see that d-branes are related to black pbranes it can be seen that a fundamental sring charged with respect to the B_mn field admit solutions somwhat similar to resissner-nodstrom black holes and these solutions are known as "black strings".

Aparently these would be similar to the d-branes but there are a few diferences. Perhaps the most interesting of them is that the d-branes can be shown not to deformate, at the firs order in perturbative calculations, the space around them (despite the fact they have mass). NS5 branes don´t share these propertie and are less addequate for "brane enginering".

A diferent kind of branes are related to M theory. In the same way that N=2 supersymmetric theroies in 10 dimensions are related to string theries one can answer if there is some fundamental theroy related to N=2 11 dimensional supergravity. A carefull analisis of the fields which appear in eleven dimensional supergravity shows that the source for them need to be extended objects (in fact one cna infere the existence of D-branes for type II strings because the 10 dimesnional supersymmetries have the same RR fields that the corresponding superstring theories to whcih they are related. Concretelly it is necessary the existence of 2 and five dimenional branes. Like they are related to M theory they are named M-branes. M theorie also appears as the S-dual of Type II A superstring (the size of the eleventh dimesnion beeing g.l where g is the string couplina nd l the string length). The M2 brane whould be associated to the fundamental string so there are not fundamental strings in M-theory.

The last type of branes I will speak about are G-strings. It can be shown that the global charges in a D-dimensional theory of gravity consist of a
momentum PM and a dual D − 5 form charge KM1...MD−5 , which is related to the
NUT charge. It is possible to construct p-branes for these charges in a very similar way that it was made for the RR gauge fields and you get a D-5 and a 9 branes which is called G-brane (gravity brane) Here D is 11 if the gravity theory comes from M-theory and 10 if it comes from supersymmetric Type II strings.

Hope that the post would be understable and that I wouldn´t have made some mistake in the exposition. Also to say that there are some other types of branes, but I think that the ones trated here are by far the most frequently found ones.

One string to rule them all...

En este journal se ha hablado mucho sobre la teoría de cuerdas, pero, sin embargo, no se ha hecho nínguna exposicion formal de la misma, Bien, es tiempo ya de ser un poco mas precisos respecto a la teoria de cuerdas,

Empezamos por lo más sencillo, explicar que es una cuerda dentro de esta teoria. Bien, en realidad es la cosa mas sencilla del mundo, una cuerda (bosónica), matemáticamente, es una curva (real) que evoluciona en el tiempo. ¿Por que alguien se preocupó de trabajar en una cuerda cómo un objeto fundamental en vez de hacerlo con partículas puntuales? La respuesta, curiosamente, es "nadie". La primera motivación para ocuparse de una teoria de cuerdas proviene de la cromodinámica cuántica, o más bien al estatus de la físca de hadrones antes de aparecer la cromodinámica cuántica. Sin entrar en muchos detalles señalar que se sabe que el neutrón y el protón, las partículas que forman el núcleo atómico no son partículas elementales, estan formadas por (3) quarks. Esos quarks se describen por una teoria gauge, la SU(3). Lo curioso es que si los quarks, y las partículas que median su interacción, los gluones, deben formar estados ligados (protones, neutrones, y en realidad todas las partículas hadrónicas) debe haber algo que impida que haya quarks libres, que nunca se han observado. Eso llevó a que en un momento dado se propusiera un modelo fenomenológico bastante descriptivo. Los quarks estaban unidos por algún tipo de cuerda, es decir, existían cuerdas que tenían un quark en cada uno de sus extremos, el confinamiento (ausencia de quarks libres) se debería a que si se estiraba demasiado esa cuerda se rompía en dos nuevas cuerdas cada una con su pareja de quarks, en realidad un quark y un antiquark, en sus extremos (para el protón o neutron era necesario tres cuerdas unidas por un extrem oentre sí y con un quark en los otros extremos). Hacia falta ponerle mates a esa idea, y es lo que se hizo allá por el 75. El problema es que esa teoria tenía un "inconveniente", en su espectro aparecía una partícula de spin 2 que claramente no encajaba en el modelo de quarks, más adelante se reinterpreta la teoria de cuerdas cómo una teoria fundamental y esa partícula de spin 2 pasa a ser el gravitón. He hblado que en el espectro de una teoria de cuerdas hay partículas, bien, esto significa, hablando de manera simplificada, que las cuerdas vibran y que cada modo de vibración se identifica con algún tipo de partícula. Según esto cada partícula conocida sería un modo de vibración de una cuerda. Como ese rango de partículas incluye los fermiones (por así decirlo las partículas que forman la "materia") y los bosones (las partículas que median las interacciones entre la materia) tenemos que la teoria de cuerdas sería una teoria que explicaría toda la física conocida, serí auna teoria unificada. Y además sólo tiene un parámetro libre, la tensión de la cuerda, así pués con la media de un sólo parámetro se tendria el valor de todos los demás parámetros de la fisica pués sería deducibles matemáticamente a partir de esa tension. Tras este previo sobre fenomenológia, no especialmente riguroso, vamos con algo de mates.


En matemáticas, geometría diferencial básica (sin usar formalismo de variedades), una curva es un lugar del espacio de dimensión uno que puede, en un sistema de coordenadas, describirse mediante una parametrizacion.



Aquí σ es el parámetro que describe la curva y las X^μ son las coordenadas. El índice μ varia desde 0 hasta D-1, dónde D es la dimension del espacio-tiempo donde se situa la cuerda. Bien, esto es una curva, una cuerda es una curva que se deja evolucionar enel tiempo, es decir, que aparte de la dependencia en σ hará una dependencia en τ (tiempo propio).




Bien, esto es la "cinemática" de la cuerda, nada particularmente complicado, pasemos a la dinámica. Cómo se ha discutido por aquí, y es bien sabido, en física la dinámica suele inferirse a través de una función lagrangiana, ¿que lagrangiana debe describir la cuerda? Bien, hay dos posibles, la más sencilla, conocida cómo la de Nambu-Goto surge de generalizar el lagrangiano de una partícula libre en relatividad especial, que recordemos es:



dónde, cómo es habitual en física, el punto sobre la coordenada denota derivacion respecto al tiempo. Esta acción representa la longitud de la línea de universo de la partícula relativista, es decir, una partícula puntual, matemáticamente un punto, al evolucionar en el espacio-tiempo describe una rayectoria, parametrizada por el tiempo τ. La acción es la longitud (en la métrica de Minkowsky) de esa curva. Pués bien, una particula al evolucinar en el tiempo describe una curva. Una curva al evolucionar en el tiempo describe una superficie, ergo la acción de Nambu-Goto de la cuerda va a ser el área (minkowskiana) de esa superficie:



Bien, esta acción es sencilla de entender, mera genralización de la acción de la partícula clásica. El problema es que aparece una raiz cuadrada, y eso, cuando se quiere proceder a tareas de cuantización, es algo muy molesto. Así pués se prefiere usar otra accion, la de Polyakov. El truco es expresar el área mediante una métrica intrínseca de la superficie, denotada por h, en concreto tenemos:



dónde la fórma concreta para h es:



Bien, esta es la forma de la acción. En mecánica clásica una vez que tenemos la acción normalmente lo siguiente que hacemos es calcular las ecuaciones de movimiento asociadas a ella (ecuaciones de Euler-Lagrange). Pero antes de hacer eso hace falta señalar unos aspectos importantes. Esta acción, cómo muchas otras que aparecen en teoria cuántica de campos, tiene simetrias, es decir, existe un grupo de transformaciones de los campos que dejan invariante la acción. La accion de Polyakov tiene tres simetrías:

(i) Invariancia Poincaré , (ii) invariancia bajo difeomorfismos de la Worldsheet , y (iii)invariancia Weyl (invariancia de escala).

Estas invarianzas se expresan matematicamente en términos del tensor energía-momento, análogo al de la relatividad general, cuya expresión es:





La invarianza bajo difeomorfismos implica que este tensor (que nos da cuenta de la energía y el momento de la cuerda) debe conservarse, es decir:



La invarianza Weyl se traduce en: .

Bien, esto concluye la breve por ahora el análisis de las simetrías, vamos a poner la ecuación de movimiento:

*

Una vez se tiene la ecuación de movimiento se debe proceder a resolverla.

Habíamos dicho que teníamos siemtrías. La invariancia de la acción bajo esas simetría se traduce en que hay grupos de soluciones equivalentes. Necesitamos un modo de deshacernos de las soluciones redundantes, eso esta relacionado con las ligaduras de las que hablé en los post de LQG. No obstante sin necesidad de saber los detalles de la teoria de ligaduras de dirac podemos entender bastantes cosas, sigamos.

Cuando queremos resolver ecuaciones diferenciales (en este caso en derivadas parciales) se imponen condiciones de contorno. En este caso estas condiciones tiene interpretación cómo condiciones en los extremos de las cuerdas, tenemos cuerdas abiertas (condiciones de Neuman) y cerradas (Diritlech).

En realidad más adelante se comprobó que había mas detalles a tener en cuenta en esto en relacion conla teoria de branas, pero no merece la pena ocuparse de ello en esta introducción.

Tenemos las condicones de contorno, vamos a proceder a encontrar soluciones a la ecuación de movimiento (*). Para hacerlo hay primero que fijar un gauge, elegimos el conocido como gague conforme ahí la ecuacion de movimiento se reduce a la ecuación de Laplace y la solución nos queda para la cuerda cerrada:





y para la cuerda abierta:





dónde son la posición y el momento del centro de masas de la cuerda.

Bien, hasta aquí lo básico, la parte clásica. Enla cuantización, que no trataré en este post, los α de las dos últimas ecuaciones se convertirán en operadores de creación/aniquilación que se corresponderían con las partículas observadas en la fisica del modelo standard. Habrá que imponer la anulacion de la derivada del tensor de energía momento lo quedará lugar a la famosa álgebra de Virasoro. Y además habrá que comprobar que las simetriás de la teoria clásica se respetan, esto no es algo precisamente trivial, todo lo contrario, esas simetrías sólo se respetan si la dimensión (conocida como dimension crítica) en que se propaga la cuerdas es distinta de 4. Aquí he estado explicando la cuerda mas sencilla posible, la cuerda bosónica; para esta cuerda la dimension crítica es 26 (25+1). En realidad la cuerda bosónica no es realista, para empezar, cómo su nombre indica, no tiene nada mas que bosnoes en su espectro. Cuerdas realistas requieren fermiones, eso implica introducir supersimetría y así entramos en el reino de las supercuerdas, par estas la dimension crítica es 10 (9+1). Desde luego hay muchísimo más que decir sobre la teoria de cuerdas, no en vano un libro de 750 páginas tiene algunos capítulos que más que un libro de texto parece un rápido review de resultados, pero creo que lo expuesto puede servir de orientación de a que nos estamos enfrentando al hablar de teoria de cuerdas.

Breviario de goemetría no conmutativa

¿Que es esto de la geometría no conmutativa? La respuesta varía un poco dependiendo a quien preguntes, un matemático o un físico. Un matemático dirá algo así cómo que es algo relacionado con la geometría algebraica y enseguida te atacará con un montón de álgebra abstracta. Un físico empezará por mencionar la mecánica cuántica (que raro xD) y te dirá algo del estilo: "El principio de incertidumbre de Heisemberg indica que los operadores posición y momento no conmutan, i.e. [x,p]= ih, esto puede verse como una incapacidad para resolver un área en el espacio de fases del orden de la longitud de planck. Bien, pues la idea de que el campo gravitatorio pueda estar cuantizado podría implicar que algo así deba aplicarse para los operadores de posición, es decir [x,y]=ih, que, análogamente a lo anterior, significa que no podemos resolver un area, esta vez enel espacio 'real' menor que la longitud de Planck".

Bien, esto para empezar, cosas así ya se le ocurrieron a Heisenberg tiempo ha, luego hay que ver como implementar esa idea. Y ahí toman contacto las ideas de físicos y matemáticos.

¿Como podemos caracterizar los puntos de una variedad? (una variedad es la descripción matemática del espacio-tiempo de la física) Bueno, cualquier geómetra algebraico dirá "por sus funciones las conoceréis". Vamos, que ellos se dedican a estudiar las características del espacio-tiempo mediante las funciones definidas en él. ¿Que características? Bien, suelen empezar por una cosa llamada cohomologia que permite estudiar característica globales de las variedades y ver cuando dos variedades son topologicamente equivalentes (pueden deformarse unas en otras de manera suave, es decir, sin romperse, bueno, al menos esto es para variedades "normales", no sé yo sí para variedades algebraicas abstractas la cosa es exactamente así). ¿Le sirve esto de algo a los físicos? Pues en principio la verdad es que no mucho. Al fin y al cabo estamos interesados en propiedades mas bien locales del espacio-tiempo, así que la cohomologia no sería una gran prioridad. Sería mas interesante ver lo que dije antes, implementar eso de que las coordenadas no conmuten.

¿Cómo usar las funciones de una variedad para que las coordenadas no conmuten? Bien, podemos caracterizar un punto de una variedad mediante los valores que toman en ese punto todas las funciones que están definidas en la variedad. Y a partir de ahí podemos definir cosas como las distancias entre punto y tal. Bueno, esta es una posibilidad, pero luego ya veré que se siguen caminos menos obvios. Pero de momento sólo he dicho que puedo caracterizar las coordenadas mediante funciones ¿cómo hago que las coordenadas no conmuten? Bien, fácil, haciendo que las funciones no conmuten, es decir, defino un producto no conmutativo de funciones, i.e.

Producto ordinario: f(x).g(x) / f.g=g.f <=> [f,g]=f.g - g.f=0

Producto no conmutativo: f(x)*g(x) / [f,g]= f*g - g* f ≠0

¿Y como defino yo productos de esos que tengan algo de lógica en física? Hay varias formas, la más sencilla es usando lo que se conoce como deformation quantization:

f*g=f.g + h/2{f,g} + O(h²)

dónde {,} es algún bilineal.

Bien, vamos a dar una forma concreta de ese producto no conmutativo. Antes puse que la motivación inicial era [x,y]=ih. En realidad debemos ser algo mas flexibles y tener algo del estilo:

[x_u,x_v]=θ_uv

El producto de funciones que nos dará esa relación para las variables en la variedad es:

(f*g)(x)= e^{iθ_uv∂_uy∂_vx}f(y)g(z)|y=z=x

Bien, vale, de acuerdo, ya tenemos un producto que nos implementa esta idea. ¿que podemos hacer con él?

Pues debemos hacer física. ¿Y que tenemos en física? Muy sencillo, tenemos que sobre la variedad hay campos, campos que son las funciones de onda que representan las partículas e interacciones fundamentales. Un momento ¿he dicho funciones? ah, pues sí, los campos son funciones, un tanto especiales por las características que implementan pero funciones después de todo, por tanto ya tenemos un modo de hacer notar a los campos físicos que viven en un mundo no conmutativo, por culpa de la gravedad cuántica y tal. Para ser exactos los campos los estudiamos mediante lagrangiano . Esos lagrangianos se forman con productos de campos (y derivadas de los mismos) Pues bien, sustituyanse los productos ordinarios de campos por productos no conmutativos y se tendrán teorías de campos no conmutativas. De hecho el producto no conmutativo puede expresarse en términos de operaciones convencionales y por tanto el efecto de la geometría no conmutativa es generar un lagrangiano diferente al habitual pero que se obtiene de él mediante este procedimiento. Pero si nos olvidamos de su origen es un lagrangiano como cualquier otro que puede proceder a meterse en algún "algoritmo de cuantización" y ver que pasa. Pues bien, este programa esta implementado para el modelo standard.

Ahora bien, las θ_uv esa ¿de donde salen? Las ponemos a mano? Ah, bueno, pero es que hemos dicho que todo esto viene de una teoría cuántica de la gravedad. Así pues las teorías de cuerdas (y tal vez la LQG) deberían tener algo que decir sobre esas θ_uv. y efectivamente, así es, al menos para las cuerdas (los de LQG esto lo tienen en pañales). Resulta que en el matrix model para la teoría M aparecen relaciones de no conmutación para las coordenadas, así que ya tenemos enlazada la geometría no conmutativa con las cuerdas . En realidad hay mas modos de enlazarlas, pero no hace falta ser exhaustivos.

Bien, he presentado el programa de la geometría no conmutativa desde un punto de vista de física (y he sido un tanto descuidado con detalles de rigor). Los matemáticos, en especial Alain Connes, implementan estas ideas en términos mas abstractos, construyendo análogos del operador de Dirac y cosas así, y introducen C* álgebras también. En el fondo son cosas equivalentes, pero cuadrar cosas entre ambos formalismos es una buena manera de asegurarse un dolor de cabeza.

En fin, estoy empezando con la NCG, así que puede que haya algún errorcillo conceptual por algún lado, si eso ya lo rectificaré cuando dé con él. Y la verdad, siendo entretenido el tema no sé si seguiré profundizando mucho en él. Eso sí, es una cosa que se supone que todo físico serio debe saber, así que no esta de más aprenderlo.