Black holes information distortion paradox

A few days ago a friend of mine, graduate theoretical physician, but not an active physician nowadays, and an ocasional reader of this blog,let me know of a new in the media versing about a resolution of the "black hole information paradox". The new was published in many webs, for example here. By the same time a thread was opened in physics phorums about the topic, concretelly Physicists Demonstrate How Information Can Escape From Black Holes based in LQG.

Ok, I suposse that I would have to somehow give an opinion about the new. I have waited a bit to see if, apart of physics phorums people, some of the big (or even not so bigs) ones on the blogosphere said something about the particular. Afther all Astekhar, the mainauthor of the paper behind the new, is one of the greatests personalities in LQG (the whole field begins with a work of him about new variables in canonical gravity) and the theme is very catchy, to say the less. For some reason there has not been such an entry in referential blogs, so I´ll try to give my humble opinion abot the particular.

First of all to say that I find that the claim of the new somewhat distort the actual nature of the achivement. Suposedly the paper solves the questions in the framework of LQG. Afther all in the news release you can read:

"Once we realized that the notion of space-time as a continuum is only an approximation of reality, it became clear to us that singularities are merely artifacts of our insistence that space-time should be described as a continuum."

The idea of discrete space-time strongly suggest that they are talking about LQG. Well, in the physics porum post somone pointed to the arxiv paper behind the news release, concretelly this, Information is Not Lost in the Evaporation of 2-dimensional Black Holes. The first bad thing comes in the title, "2-dimensional black holes". That is they solve the problem in a simplified modell, that always opens the possibilitie that the problem could not be solved in the full environment, afther all 3-d quantum gravity is very diferent from 4-d one.

Anyway, let´s see what is going on. In the last post I talked about LQG and I did a brief description of how LQG treats black holes (or at least one of the ways they do it when they face the singularity problem). As not every reader of this blog is assumed to speak spanish I´ll re-explain it. They don´t work in the full LQG framework but in modell with reduced simmetry. They get the Scharschild solution (a solution for vacuum Einstei equations, statif and with radial symmetry) and write the hamiltonial constraint equation fo it. They treat the radious as a discrete time coordinate. That results in a diference equation that can be solved and they show that they can evolve the solution for negative values of the radious, so they, seemengly, advoid the central singularity. Before reading the paper of Astekharet all I tried to figure how they could have procceded. To begin with the information paradox problem is related to matter in the vecinity of the horizont. So they would need to introduce in the description mttr in some wayor another. The original work of Hawkings that raised the whole problem used a fiexed Schwarschild background and an scalar field propagating in it. By the properties of quantization of fields in a courved backgrounds it was known that a vacuum state contianing no particles for aan observer is transformed in a state containg particles by a bogoliougov transformation for another non inertial observer. Playing with that, and with the conformal diagrams of black holes, Hawking derived that black holes actually emit radiation, in thermal quilibrium. That raises the problem that the black hole aan evaaporate because that procces. But the b-h was formed by matter in a pure state, and the thermally described matter is in a mixed stte, so the evolution would be not unitary (that is a bried description of the problem we are trating, for the ssafe of someone wouldn´t know it). Canonical LQG, the one in which Astekhar usually works, normally trates pure gravity, althought it can describe non fermionic matter also. Knowing that I thoguht that they would use some variant of the singularity removal approach including a klein-gordon field. Well, I was too naive.

They work in something called "Callen-Giddings-Harvey-Strominger (CGHS) black holes". I had not previous knowledge of that model, but the names behind it sound me "stringy", in particular Strominger is mainly an string theorist. Well, I was not wrong this time. The paper makes begins with a brief description of the hawking problems, some previous aproachs to the solution (açone by Hawking himself aaproach based in the maldacnena AdS/CFT corresondence) and just afther that talks about some workd in the early ninties triying to solve it in a toy two dimensional modell, the CGHS.

Just before writing the actual equation of the model the aouthor make a very courious advise:

"Although our
considerations are motivated by loop quantum gravity, in
this Letter we will use the more familiar Fock quantization
since the main argument is rather general"

So they say that we are not going to see a formalism related to LQG, alathought LQG is behing the scene. Well, that means taht we must belief in LQG, but we are not ging to see it. Ok, lets belive, at least for a while. Let´s see (part of)the lagrangian describing the model:

...

Now it is when one can beguin to be really surprised. We have that Phi is said to be a dilaton. But a dilaton is a field related to string theory. All of the strings theories have a dilaton. So we are in a modell inspired by string theory (an aspect that it is not mentionesd anywhere in the paper). R is the curvature and f is an scalar field. Well, ok, no problem, someone would expect their appearence.

Afther that they introduce the equatios associated to the lagrangian and begins the task of finding solutons resembling a black hole suited for their purposes. First they affront the classical case. They do it in a perturbative, recoursive, way. That is, they choose a candidate metric, calculate the stress tensor for the fields and reintroduce it in the Einstein equation. By dong that they find that the metric an develope a singularity that they can identify as a proper black hole.

Afther that they consider a quantized version, they add hats xD). Not, serously, they use a fock space tratement (in teh spirit of the Wald aproach to quantization in courved backgrounds, but this time quantizing the metric also). They afrront the uestion of quantization (solving the conmutator eqations to say that) by a bootstrapping procedure, a recoursive way similar to the classical one. They do the suual stuff of identifiying the average values with the classical solutions,but they face a problem when the metric becomes singular, and they cann not continuate the bootstraping. Afther that they use another procedure, a mean field approach MFA. They argue that the e relevant part to solve the information paradox depends on the behaviour of the MFA in the near future ifinity and with some 3 extra sumptions ( they explain that two of them aare commonly accepted and that the other is very natural) they can calculate the S-Matrix and se that it is unitary.

Well, the detaills of how valids are the asumptions (2-D space time, MFA, asymtotic regions, etc) is something that, fourtunately, I don´t need to judge. The key point that I want to raise is that what we see in the paper is very, very, far from any formalisms related to LQG. So to claim that this can be seem as a trioumph of LQG, if they don´t bring a future a paper (or smewhat point me that I am missing something important) where they addapt the calculations to something more LQG like, is to somewhat distort the truth. Or, at least, a too propagandistic deformation of facts ;-).

P.S. Seriously, I would like to leave this tasks to the famous physicists bloggers. For example, I am still wating Sean Carrol to post about the 't Hoof paper I writted about two posts above.

The trouble with LQG

A través del excelente blog física en la ciencia ficción llegué a otro blog, la bella teoria. Supongo, no lo sé con certea, que eso de "la bella teoria" debería ir por la teoria de cuerdas, que en su momento alguna gente consideró matemáticamente bella y elegante. En todo caso en el blog ví que se trataban con cierta frecuencia temas sobre teoria de cuerdas (aunque no exclusivamente). También observé que algunas de esas entradas contenían algunos errores elementles. De hecho le señalé al autor uno en una de las entradas, a lo que me respondio dándome las gracias por el aviso. Posteriormente he visto algún que otro error, pero no me he molestado en írselos indicando uno a uno. Yo no me considero una autoridad en teoria de cuerdas ¿alguien aparte de Lubos Motl se atreve a asumir ese rol xD? y soy consiciente de que es probable que en este blog haya errores en algún que otro aspecto. Además el tono de ese blog es claramente divulgativo, para un público general y las entradas estan escritas de manera amena y cuidda (con fotos y cosas así). Dado que no hay en español muchos blogs (si es que hay algún otro) que cumpla similar función (yo, por ejemplo, oriento este blog a gente de un nivel de conocimientos bastante mas elevado) no tiene sentido ir señalando fallitos, que es algo que supongo podriá resultar irritante.

El caso es que el autor ahora ha ido a leer el polémico libro de Lee Smollin "the trouble with physics". Tras hacer una presentación del mismo en una entrada publico una segunda titulada Gravedad cuántica, continuando la revolución de Einstein Cómo quiera que es muy poco probable que la gente que lea su blog (y aparentemetne el mismo autor) este al tanto de los detalles de las "string wars" en los blogs Motl ,Distler, String Coffe y not even wrong, y de las críticas que los teoricos de cuerdas han elaborado sobre el estado actual de la LQG dejé una respuesta, que no ha recibido ulterior réplica dónde resumía algunas de ellas. Por su interés para los lectores de este blog dejo aquí esa respuesta:

"Salvador, lo que estas comentando es lo que se conoce cómo LQG. loop quantum gravity, o gravedad cuántica de lazos en español.

Dado que en tu anterior post hablabas del libro "the trouble with physics" asummo que has sacado de allí la información sobre este post. Aprovecho pués para comentarte ambos temas en esta respuesta.

Llevo siguiendo la LQG, a nivel técnico, no de divulgacion, desde el 2003, cuando se hizo populara en una hábil maniobra publicitria de algunos de su máximos represetantes. Fué un producto muy bien vendido, sobre todo mediante un paper de review de Robert Thieman. Te presentan la idea general, explicando cómo el programa de cuantización canónica lleva a que el operador área esta cuantizado. Luego hay argumentaciones de que, mediante algo conocido cómo "double special relativity", que extiende el formalismo de la relatividad especial al caso de una longitud privilegiada (la de Planck) se llega a que la velocidad de la luz en el vacio depende de la frecuencia, para luego señalar que los satelites GLAST podrian llegar a comprobar eso en un lapso de tiempo breve (su lanzamiento estaba previsto para el 2005).

De ahí pasan a argumentar que deducen la entropia de un agujero negro, reproduciendo la fórmula conocida mediante la aproximación semiclásica de que depende de una potencia del área. No entran en los detalles, que dependen de una construccion de relatividad general poco conocida llamada "isolated horizonts" (y también de el concepto relacionado de "dynamic horizonts". Cómo casi nadie esta familiarizado con eso conceptos consiguen que en una primera lectura uno no caiga en que en el cálculo se esta asumiendo, como punto de partida, que la entroia depende del área. Además la teoria de la LQG tiene un parámetro libre, concido cómo parámetro de inmirizzi. Ajustando ese valor se consigue que la fórmula obtenida se ajuste. Total, que el único punto medianamente serio del cálculo es que el área esta cuantizada qu ehay que dividir el área total entre la unidad mínima de área para encontrar el número posible de microestados.

Luego pasan a cosmologia. Anuncian qu econsiguen demostrar el "rebote", es decir que no puede haber un big crunch pués usando factores cuánticos se comprueba que el colapso se detiene en un radio mínimo y luego hay un nuevo big bang. Claaro, eso no se hace dentro de la teoria completa sino restringiéndose a "minisuperespacios", es decir, subgrupos de las métricas posibles. Vale, se admite que es una aproximación, el problema es que no dan ningún criterio que permita hacerse una idea de cuan buena o mala es esa aproximación.

Por cierto, la LQG canónica, basada directamente en la gravedad clásica de Einstein y luego cuantizada de un modo raro, presuntamente no perturbativo, es estática. Para tener un espaciotiempo dinámico tiene que irse a un formalismo lagrangiano (las spinfoams). El caso es que las spin-foams no parten de la relatividad de Einstein sino de unas teorias clásicas que con ciertas restricciones, son más o menos clasicamente equivalentes a la relatividad de Einstein. Además hay varias de ess teorias clásicas y cada una lleva a una teoría cuántica esencialmente diferente.

Y bueo esas dos son las líneas fundamentales, pero aparte hay otras que asumen de partida que el espaciotiempo es discreto, lo cuál es mucho suponer. Y no quda muy claro la relación entre ellas.

Mas importante es que esas teorias cuánticas deberían tener un limite clásico que reprodujera la gravedad clásica de Einstein. Pué sbien, no han conseguido eso de una manera medianamente clara (lo mejor que tiene es obtener, mediante una serie de aproximaciones de dudosa justificación) reproducir, en el marco de algunas spinfoams algo que debería ser similar al potencial de Newton.

Vamos, que no digo yo que la LQG sea un sinsentido, pero desde luego tiene muchos problemas bastante serios.

Respecto a su mejor punto, la posibilidad de confirmación de la dispersión de la luz en el vacio, al final los satélites GLAST han tardado muchísimo en lanzarse, (deben estar a punto de salir ahí fuera por estas fechas). Un experimento en tierra hace unos meses dió un resultado preliminar en esa línea, pero no es conluyente (y hay ciertas teorias de cuerdas, bastante raras todo sea dicho, las cuerdas de liouville, que predicen algo similaar. De hehco fué un fisico de cuerdas quien hizo populñaar ese experimento).

Y, por si no esta claro, la teoria de cuerdas también es una gravedad cuántica, así que no se puede presentar esto cómo si fuera le más de lo más en gravedad cuántica y dejar de lado la teoría de cuerdas.

En fín, no me parece mal que comentes sobre la LQG, pero cuidado, que no es oro todo lo que reluce con es gente."

Bien, tras esa entrada Salva ha publicado otra . Más allá de los agujeros negros. No he leido "the trouble with hysics, sólo los comentarios, unos favorables (en la comunidad LQG) otros adversos, así que no tengo la certeza, pero si la firme sospecha, de que los datos de esa entrada salen de ese libro. En cualquier caso lo que si conozco es el tema específico tratado (agujeros negros en LQG) pués lei en su momento unos cuantos artículos originales sobre el particular. La entrada en sí no contiene ningún error, pero creo que es interesante que complete la respuesta que reproduje antes para explicar aalgo respecto a la singularidad de los agujeros negros en LQG.

El tratamiento de los mismos se hace inspiróandose en el formalismo de la LQG canónica, es decir, partiendo del formalismo Hamiltoniano. Al igual que se hacía en cosmologia con los modelos de Friedman-Robertson-Walker* se elige un subgrupo de métricas, para empezar la métrica de Schwarschild, que describe el campo gravitatorio estático y con simetría eférica que se supone es producido por una distribucion de masa esférica, idealmente concentrada en un punto, y se construye el hamiltoniano que corresonde a esa métrica. Recuérdese que en gravitación canónica el hamiltoniano consta exclusivamente de lo que se concoe cómo ligaduras (una ligadura es una relación que surge, en sistemas denominaods singulares, debido a que no es posible invertir la transformación que nos daría los "momentos" en términos de las "velocidades"). Tenemos pués el mismo problema que citaba antes para la LQC (loop quantum cosmology), que estamos eliminando grados de libertad , por estar restringiéndonos a un subconjunto de métricas, haciendo por tanto una aproximación. Lo malo es que no se tiene una estimación del error cometido en esa aproximación. La gente de teoria de cuerdas arguye que se estan eliminado demasiados grados de libertad y que la aproximación es totalmente injustificada.

Hay, sin embargo, un aspecto en que el tratamiento de los agujeros negros difiere de la LQC. En FRW tenemos una variable temporal, en una solucion de agujero negro tipo Schwarschild no. La forma de proceder consiste en que la ligadura cinemática de la LQG puede resolver en términos de Spin Networks. HAy una cantidad infinita, pero numerable, de spin networks. Bien, la "evolución temporal", es en cierto modo, la accion del hamiltonianao sobre la base de esas spin networks. La ligadura toma la forma de ecuacione en diferencias dónde el tiemo es el índice en la relacion de recurrencias. Para ese caso particular el índice de las spin networks pasa a ser el radio, es decir, el radio juega el papel del tiempo. No estoy 100% convencido de que me crea del todo esa interpretación, pero admito que tiene un cierto ingenio. Además en gravedad clásica es bien sabido qeu el radio si juega el papel del tiempo (por el cambio de signo de los componentes de la métrica en el interiro del horizonte de sucesos dle agujero negro). En todo caso lo que se hace es resolver esa ecuación en diferencias que da la evolución. A diferencia de la geodésica de la relatividad geneal clásica, que termina en el centro del agujero negro en un tiempo propio finito, o´nde el vlaor de la métrica se hace infinito (es una singularidad auténtica, y no producto de la elección de un mal sistema de coordenadas, como la dle horizonte de sucesos). Esa evolución puede prolongarse para valores negativos del radio. La interpretación de ese resultado es algo sobre lo que en los artículos no habia una pronunciacion clara, pero se aarguía que podria ser similar a los "baby universes" que popularizó Hawkings. Esos "baby universes" surgíeron en la aproximacion euclidea a la gravedad cuántica (el formalismo favorito de Hawkings). Alli lo que se hace es trabajar con métricas euclideas (resultado de cambiar en la métrica de Einstein de tiempo real a tiempo imaginario). En esas teorias también se puede eludir la singularidad central yéndose aun tiempo imaginario, eso se interpreta como la creación de un "baby universe". No conozco mucho la gravitacio cuántica euclidea, pero creo que esos resultados se conseguían alli haciendo una aproximación tipo WKB, con lo cuál no eran del todo firmes. De hecho yo siempre había visto la LQG cómo una versión de la gravitacion cuántica euclidea en un formalismo mas matemáticamente preciso, bastante cercana a su espíritu (algo con lo que no coinciden los teoricos de cuerdas). En cualquier caso esa es la secuencia de razonamientos que llevana evadir la singularidad del centro de un agujero negro en LQG. La fecha de publicación del libro de Smollin coincide más o menos con la fecha de los artículos que leí en su momento, con lo cuál no recoge los desrrollos posteriores de ese trabajo. La verdad es que esos trabajos en agujeros negros fueron casi lo último que leí con detalle de LQG, y que desde entonces la he seguido muy por encima, a través de los posts de physic forums (excepcion hehca del trabajo de Reuters en el grupo de renormalizacion, que sigo teniendo pendiente exponer por aquí algún dia de estos). Tengo entendido que lo que se hizo fué generalizar los resultados a conjuntos más amplios demétricas (especialmente en LQC, aunque también algo se hizo en agujeros negros) para ver si en esos conjuntosmás ampliso se mantenian los resltados y que estos o eran debidos a restringirse a métricas con excesiva simetría. Aparentemte se mantenian parte, la parte esencial, de los resultaddos, pero ya no puedo dar detalles (si hay algún experto en LQG entre los lectores se agradecerían las posibles puntualizaciones).

En fín, sirva este posts, en español, cómo rsumen de algunas objeciones comunes a la LQG expuestas de una manera que aunque somera, tenga un mínimo de detalle

Gerard ’t Hooft: A diferent string theory

The first oe paper is LOCALLY FINITE MODEL FOR GRAVITY written by Gerard ’t Hooft. If by some casual you dont know who ´t hoof is just to say that he has a nobel prize by proving that the gauge theories, wich are the basic ingredient, of the standard model, are renormalizable. A few physicists I know consider him the last greater phyisicist (Steven Weinberg would be considered slighly earlier in time). ON the other hand be sure that Lubos Motl is not among his fans ;-). Ok, let´s leave sociology and go into physic. The abstract of the paper is this:

Matter interacting classically with gravity in 3+1 dimensions usually gives
rise to a continuum of degrees of freedom, so that, in any attempt to quantize
the theory, ultraviolet divergences are nearly inevitable. Here, we investigate
matter of a form that only displays a finite number of degrees of freedom
in compact sections of space-time. In finite domains, one has only exact,
analytic solutions. This is achieved by limiting ourselves to straight pieces of
string, surrounded by locally flat sections of space-time. Globally, however,
the model is not finite, because solutions tend to generate infinite fractals.
The model is not (yet) quantized, but could serve as an interesting setting
for analytical approaches to classical general relativity, as well as a possible
stepping stone for quantum models. Details of its properties are explained,
but some problems remain unsolved, such as a complete description of the
most violent interactions, which can become quite complex.

The paper begins with some considerations about 2+1 dimensions and the role of pont partile matter as source of curvature, in the form of a wedge in space time. Inspired by that view he considers the extension of this to 3+1 dimensions. The role of the point particles are now strings. Why? Simply because the aditional dimension is perpendicular to the others and so a point becomes an infinite string. In principle it could look a bit arbitrary, and not general. But the idea seems to consider the space-time sourruounding that infinite strings.

He writes the energy momentun tensor for that strings (well known for people aware of cosmic strings). Later he considers moving and interacting strings. In considering this he concerns about holonomy so maybe the reader would consult something about this topic if he does´nt know it previously. The wikipedia entry is specially good about the topic so it could be a quick start guide. The very quick idea of homotopy is to consider a map betwen parallel transport of vectors around a closed curve and the group associated to the bundle (wich defines the very concept of paralell transpoort).

Afther that he considers interactions of strings. The first claim is as a resoult of interactions, connections, he cant´consider infinite strings alone anymore. Diferent types of collisions are analized. He cnsiders vaious posibilities and takes care about some possible issues (for example, rotating strings would create spacetimes with closed timelike curves as is well known since the work of Gott). I am not sure of how much of this work would intersect with the well stablished resoults about networks of cosmic strings (gauge or superstring ones). It would be fine if some reader would know it and could say something about it.

The final conclusion he claims is that he can get all the degrees of freedom of gravity by pieces of straight strings. In this way he could study gravity just from this. Seemengly this is somewhat similar to Regge calculus (a discretized aproach to quantum gravity)using strings instead of points in the nodes. He also says that in this sense is just the opposite to some papers triying to get matter from ure gravity (in a clear reference to Smollin program in the octopy). In the paper a quantizaion of the model is not made, that is announced for a future paper. About that paper he says that he will not follow traditional quantiztion proceduers based in lagrangian mechanisms,partially because the model seems not to admit a Lagrangian formulation). AS an advance a claim is made that the theory will not have ultraviolete divergences but possibly will have problems with infrared regime.

Well, I still have to re-read carefully some pieces of the paper, and a definite juice will only be possible when the paper on quantization would be available. Also it would be interesting to see how it is recived by the mainstream physic comunity. For example Sean carroll has announced that he will speak about the paper (it is how I knew about it´s existence). And, I guess that also Lubos Motl will have something to say, given his aparent animosity against t' hoof (maybe because he has sometimes soped favourably about the LQG comunity). Personally I consider ’t Hooft a very interisting figure and I like to be aware of what he does, even if I don´t necesarilly agree with all his conclusions.

New, complementary, blog

As announced I have created an aditional blog inwordpress. The actual url is http://freelancescience.wordpress.com/.

The idea is to keep this blog for quantum gravity related stuf and the other for diferent aspects of physics, math, and, ocasionally, another sciences. A separate interest is to see how well it works the LaTeX functionality of wordpress. If it works fine, and blogspot refuses to allow LaTeX, I would consider the posibility to migrate this blog there (afther all they have facilities to do so).

Advise, the other blog will have most of their entries in spanish, because of the expected target audience, if some non spanish speaker is particularly interested in seeing english entries he could let me a message and I would see what it can be done.

Fresh air for string theory

The very recent mounths seem to have brought great news for string theory. I´ll write in this post a brief guide to the relevant papers.

The first, cronologically, is this paper by Beasley, Heckman and Vafa. It is a paper where, for the first time, it is addressed the task of constructing phenomenology from F-theory. A second one is announced where the program started in thiw will be concluded. If 125 pages is more time of what you have available just now you can try to read a sinopsis in the Jackes Distler glog, concretelly here. Lubos also wrote his own comment, try to search for it in his blog ;-).

Maybe the reader has no too much familiarity with compactifications and similars. In that case, or even if he has, it would be recomendable the following paper Les houches lectures on constructing string vacua. The great point of that paper is that it covers in a single paper approach for many types of strings, and many types of vacua (compactifications, fluxes, etc).

Another very interesting topic are advances in M-theory and their interactions. A review explaing the topic can read here. Well, two papers have changed things and some people (yes, you are rgiht, Lubos) has told that it may be the beguining of a third string theory revolution. The papers in question are these:Gauge Symmetry and supersymmetry of Multiple M2-Branes and M2 to D2. You can read a brief description in Lubo´s or Distler´s blogs. I particularly recomend, if you are in a hurry, the one in Lubos blog because it is easier to understand.

Not enought reading? Well, there is a very recent paper in perturbabtive string theory. Concretely some where the four loop amplitude for string theory is explictely calculated. The author of the paper is Samuel Grushevsky and the actual paper is this

For the string heater readers I would recomend reading this paper by Smollin and all. They continuate their "ocutopussy program" where the try to get the standar model from spin networks of pure gravity. To be honnest, I don´t really think that approach is particularly viable, measured by the own standards of the LQG commnuity, but it is up the readers own judice to decide what is interesting or not.

To conclud simply to say that at last I found a great reference for the basics of the renormalization group, as well as in many other topics,the three volumes book of Steven Weinberg in quantum field theory. I had never tried to read that book because of it´s extension (and because, hey, I already had a reasonably good knowledge of the subject) but I must say that despite of ít´s extension it is so well written that one can read it fast. I hope to write soon a post about the subject of renormalization group theory, that could serve as an introduction to those who would want to follow the Martin reuter papers on the subject and the discusions on Distler blog on the particular.

Status of the blog

I have writen relatively few (here and in forums) in the las times. This doesn´t mean that I would have somewhatleft the physici, quite on the contrary.

Cronologicaly the first reason to stop me writng was to do a sistematic reading of Jackes Distler´s blog. That gave me a partial idea of what had been hapening in string theory and quantum gravity in general in the very last years. One of the things that one can learn there are that some of the discusions betwen the LQG comunity and the string comunity come from online discusions. Particularly it was interesting, concerning this "string wars" a post with around 100 answers about chirality in LQG. AS a consecuence of that reading I studied the chapter on chirality of he book "topology and quantum field theory" together with all the preliminars required for it (Sheave comology, some basic K theory and Atiyha singer theorem explianed in that framework, certinly not trivial things). I had readed some chapters of that book previously, but I practically had to reread them, together with some ampliations to get (aprt of the above mentioned) a, still far for finished, understanding of anlgebraic geometry. In fact that chapter, in my opinion, is far for complete, and maybe I will read in the future some of the available reviews. Anyway I think I got a decent idea of the arguments of Distler against the Smollin arguments. In fact i guess that a better defense could have been made of the case for LQG. As I understand the problem the reason why you can´t have chiral matter in LQG is that LQG is thinked by string people as some kind of latice theory. But in latice theries you can´t have chiral matter because of the period doulbing problem (there are many places to study an introduction to QFT on the latice. Perhaps my favourite one is the chapter of the Michio Kaku book on QFT). But the thing is that, as far as I undertand it the way the spin network is thinked (or wishefull thinked, who knows?)in LQG isn´t exactly a latice theory. In particular there is no topology in spin networks, or spinfoams for what matters, (LQG is a pre-topology theory)and It would evade the topological character whch make chirality a deep nature beyond their perturbative appearence in the famous triangle Feynman graph. But, certainly, if the LQG cmunity didn´t did that defense it probably means that I am loosing some point, that is, they are the experts, I have a reasonable knowledge of canonical LQG and, to a lesser extent, sin foams, but for sure I am not an expert.

Another thing that I learned, about the string wars, in Distler´s blog is that prt of it happened in the, sadly stopped, string coffee blog. I have the intention of reading it sistemathicallly also, but some things prevented me form doing it. For example reading comments about the Lisi´s E(8) theory I realized that what I had been teached about group theory in the course at the university was far from enought. I still think, as I expresed before that group theory aalone isn´t going to give answers to quntum gravity problems. But anyway It was obvious that I needed to learn better the subject. It was a very, very ugly task. As I have said in this blog my mathemathical formation is s mathemathician, rather than as phyisician, so for me the books of group theory for phyisic are somewht like a nightmare. In mathemathics a Lie group is a relatively easy thing to understand once you understand geomtery in manifolds. The definitions are elegant and natural. In physics the idea of continous groups, in the sense of calculus, and matrix groups seems rogught, and maybe even limited. But if at the level of the group the discrepances are relatively solvables at the level of the Lie algebra the problems row fast and it is almost an act of faith to belive that definitions ocf Cartan subalgebras, root vectors and almost everything are the same as stated in, for example, the book of Georgy (Lie algebras in particle physics) and in, for example, the book of Sattinger and Weaver (symmetry groups, geometry and physics, if I don´t remmeber bad the title) where they are introduced using notions of abstract álgebra (solvable ideals and thngs like that). An added problem is that mathemathicians interest in Lie groups seems to be the clasification of symmetric spaces more than in particle physic (although they also cover it, mainly the "wight fold way). Well, anyway I, at last, learned properly about the relation betwen ral and complex forms of a lie aalgebra and it´s consequences. One added problem with group theory, as teached by mathemathicians, is that they make a good cover of the clasification of Lie algebras, and give a quick tour in representation theory (including sinor representations of some algebras). But htey uses very, relatively, asic techniches. A physicans book, on the contrary, gives an in deep tratement on SU(n) wiht basic, as well as tensor methods and the younng Tableaux technicke. Those last ones, in particular, resoult that are also used in the representations of the Lorentz group, because of the litle grou and all that. Young Tableaux are not particularly difficoult to understand, basically a way to represent symmetric and antisymmetric part of a tensor products. So, when I found then in string theory books as a way to represent the particle content of the strings, I understood what was going on, but lacking calculational confidence (It was not teached in my course at the university and I just had readed previously the basic ideas) always maade me feel that that calculation of the string spectrum was "goup theory maguffery". I still think that it´s is not the best way to show the physic content of the string theories so I recmend to read the corresponding chapters in the books of Polyakov and Zweibach where one can get a more deep physical idea of what´s going on.

Another ugly part of group theory, this time restricted to physicians oriented books, is the choice of examples. Some books, for example the one of Miller, makes an extensive use of examples extracted from non relativistic quantum theory. Anthoer´s, the one of Georgy, focuses more on particle physics. In fact I can´t say for sure that it is a bad thin, but one can get lost with so many "phenomenology" and loose the common points. A separate problem is the Poincaré and Lorentz groups. I still have not totally clear how important is to care about irreducible representations, which are necesarilly infinite dimensinal, and why more aspects are done with finite dimensinal ones. Maybe the lecture of the techniche of the induced representation, whch I still didn´t do, clarify me some things. I also have no clear why exactly are important in general the casimir operators of a representation. And I still have a vague idea of the role of chaaracters of representaions for Lie groups. AS the reader can deduce group theory has many aspects and it is easy to get a false idea tht some knows properly it. Fourtounately I never have felt that a not perfect understanding of some particular aspects of group theroy forbids me to understand the ideas of physic. In fact one thing that always had intrigued me, the way quarks were assigned charge was not rellated (as I thought) to conserved charges ia the Noether theorem but comes from the pauli principle (for quantum numbers difernet from spin) which dictates some prticular choice of the representation (assoiciated vector bundle in the language of geometry of fiber bundles).

But the previous things have not been my main, and more difficoult concern, these last times. Afther all my basic in math is solid (or at lest I thnk so) and, beyond the problem of tradution betwen pure math texts and physicans math texts I had not deep problmes of understanding. The thing which more problems gave tome is the renormalization group. I haad previously mentioned an entry of Distler about the modern renormalization group and it´s relation to the try of Reuters to find a non perturbative way to get a quantum gravity along the more traditinal lines of QFT. Well, Distler, and also Lubos, gave some ideas of what´s was ging there, and remarked how important it was to know the exact renormalization group equations. In fact Distler did recently two new posts about the topic. Well, that has suposed a big problem for me. I knew reasonably well the old perturbabative renormalization and the renormalizaation group of Callan-Symanzisk, and it´s role for the calculation of the running coupling constants. I also could get an intuitive idea of what is a releant, irrelevant or marginal operator. Afhter all similar cncepts are used in conformal field theory. But one thing is to have a vague idea aand another a proper understanding, so I went to the string wiki and pursued the review articles. And I got totally lost. The ultimate reason for that is that that ideas of renormalization group come from condensed matter (il.e. statiticall physic). And that is a very bad new for me. My knowledge of termodinamics and statistical physic was, well, er...average ;-). I mean, I had a right understanding of the microcanonical ensemble, whch allowed me to undersandthe meaning of entropy. I understod the role of the other ensambles and that you could get termodynamics from statisticall mechanics (and that point not quite well). The problem is that I never had understood the utility of thermodinamics (in my first contact with it was tacitally assumed that I already knew it and the focous was in it´s relation to statisticall mechanics, pitty tht I never had been teached it). Well, no problem, what really was needed was to learn statistichal mechanics, and to calculate partition functions, classical and quantum. I got used to learn about fery and bose statistic and, to be honest, not too much more. Beeing a "pure theoretic" that never worried me too much. I could, more or less follow the basic ideas I needed to u nderstand in solid state physic and I neveer cared too much. In fact I gained some better understanding of some aspectos of termodinamics, including a somewhat non stndar aspects, Ossanger relations, reading about thermodinamics in biologic procces. I also had some very vague notios about phase transitions, in the Erenfest classification.

Well, all of that tottally insuifient. Once I realized that by reading the availabe reviews I was going to nowere I decided to follow the ling way and to relearn all the thermodinamcic and statistical l hysic from the beguining. Afther that I readed the chapters of the Kerson Huang book on phase transition and renormalization group. There I learned about things such like "kadanof blocking", the meaning of fixed points and all that. But still I felt thaat I was lacking many detaills. I tried to read agian some reviews and I got more ideas, but still not eonguht. I learned that, beyond the work of Wilson, there were tow "exact renormalization group equations". The Wegner-Hougthon and the Polchinsky ones. I even tried to read the original article of Polchinsky and besides ewin advised that he was able to proof the renormalizability of the interacting sclar theory without using topology of Feynman graphs and the Weinberg therem I didn´t understand anything. Somewhat desesperated I readed the nobel price acceptance article of Wilson and I got a better idea of what it would be the path to follow. I went for a book of renormalization group and phase transitions for condensed matter phisicans. Concretely I got "lectures on phase transitions and the renormalization group" by Nigel Goldenfeld. AS I had readed in the Huang book the basic ideas of phase transitions I went directlly to the chapters on the renormalization group. I didn´t understand all the points, particularly I got a bit lost in the 10th chapter abut anomalous dimensions. But, in general, I got, at last, a felling that I am in the right waay. A problem (probably the only one in a very well written and clear book) is that it doesn´t use field theoretic (i.e. path integral) technckes. For that particular are recomended tow books, one of Collins "renormaliztion" and one form Zin-Justin "renormalization group and critical phenomena". To be honnest, afhter all this statistichal mechanics I was dissapointed to have to read just another book. So I tried to read agian one of the review articles, and this time, at last, I understood the basic ideas, and some of the detaills. Also I have beguined to catch the detaill of the relations betwen the old and new renormalization group equations. Not surprsingly I learned that there were some diferences in the aspects of the renormalization group that interest to condensed matter physics (local ones, basically to calculate critical exponents) and the aspects usefull for an particle physis (the so called global renormalization group). Althought seemenly not essenciall I decided to read the first chapters of the goldenfeld book in general phase transition theory. I must say that I am finding it a very good idea because It clarfies the concepts a lot better that the Kerson Huang´s book. Also, beeing so well writen, seems not to be a very mcuh time consuming task.

Afther that I plain to read a book (fortunately short) about conformall field theory oriented minlly to statisticall physic. I understand CFT as applied to string theory, but I guess that reading that book I am going to get a cleare ideas of many aspects, which I now understand at the formal levl, but, probably, have some subleties that I am missing now.

And while I passed all this time triying to fill some gaps Mr Lubos Motl has recommeded as "imprescindible" not one, but two articles in string theory, of 100+ pages any. And one of the articles is just the first part of another (probalby of similar size). Er, fine, it´s good to do quantum gravity, isn´t it? xD.

Anyway, if there is out there some lector who find that my actual publication rate in this blog is not fast enought I have good news for him/her. I have another journal, in livejournal, where I have published about other topics than quantum gravity. The level, and thematic, of that journal is to wide, it includes music, cinema, sci-fi and some more topics. Sitll it is mainly a physics/mathematics journal and there are some not too bad posts in this areas. I have decided to open anthoer blog (maybe in wordpress) where I will collect the better of that articles and post new ones. I prefer to rserve this blog exclusively for quantum gravity, and I guess the reader interested in physics and maths will be glad not to have to read posts of topic triying to search interesting things (I personally find annoying to read physics journals where most of the posts are not related to physic). I am sad to say, for english readers, that in that blog all, or almost all, entries will be in spanish, sorry for te inconvenience.

P.S. I have seen just now an answer in the last post that I had missed, I´ll try to answer it as son as possible.

¿Por que el universo tiene 3 + 1 dimensiones? (Parte 1)

Esta pregunta tiene una respuesta fácil, porque experimentalmente es lo que se observa.

La problemática surge dentro del marco de las teorias de cuerdas, que tienen cómo requisito (salvo en versiones harto polémicas cómo las cuerdas de Liouville u otras también bastante discutibles, cómo las cuerdas supercríticas) que esten formuladas en 10 (u 11 para la teoria M) dimensiones.

Aquí he hablado de soluciones "ad-hoc", cómo la compactificación de las dimensiones extra. También he mencionado las soluciones basadas en "warped geometries", he discutido con más detalle (eso sí, en inglés, por eso en este post repito cosas que he explicado antes en inglés, por si algun lector no conociese ese idioma) en el contexto de la teoria de Horava-Witten, o teoria M-heterótica. Este mes la autora de la teoria de los warped universes (junto a Kunrum Sumdrum) Lisa Randall tiene un nuevo paper, esta vez en colaboración con Andreas Karch, que pretende explicar de una manera "nautral" cómo el universo puede haber llegado a una configuración en las que sólo 3 de las 9 dimensiones espaciales tienen un tamaño macroscópico.

Para entender las argumentaciones que hace primero debo explicar algunas cosas básicaas de teoria de cuerdas. Aparte de los objetos más fundamentales, las cuerdas, la teoria requiere la posibilidad de que existan otros objetos extensos, las p-branas (p es la dimension del objeto extenso). Una 1-brana sería un objeto de dimensión 1, geometricamente una curva (real, nada de complejos, sí en algun momento usara dimensiones complejas lo indicaría explicitamente, por defecto debe entenderse siempre que estoy en el cuerpo de los reales), por ejemplo una cuerda sería un caso particular de 1 brana. Una 2 -brana sería geométricamente una superficie, El caso p>2 puede ser algo chocante para la gente sin formación matemática, pero realmente no tiene gran misterio, una p-brana sería geométricamente lo que técnicamente se conoce cómo una variedad de dimensión p, que son objetos matemáticos que generalizan las curvas y las superficies.

Las p-branas, para un p dado, pueden a su vez ser de diferente tipos (Dp-branas. gravity p-brnas, o g branas y unos cuantos casos más, el lector interesado puede buscar en el blog una discusión más detallada de los diversos tipos) De lejos las más habituales son las Dp-branas. La d viene de Diritlech, y la p indica su dimensión. La forma más simple de entender una Dp-Brana es verla cómo una región del espacio en la que pueden terminar los extremos de una cuerda abierta. Esos extremos pueden moverse libremente por la Dp-brana, pero no pueden abandonarla. Para cuerdas cerradas curvas cerradas) la definicion de Dp-brana es algo más delicada y no daré los detalles aquí de cómo se hace.

Las Dp-branas, geometricamente, no se supone que puedan ser una superficie arbitraria. El tipo más normal de Dp-brana es una superficie plana de extensión infinita (o al menos tan grande como el espacio disponible). Esto contrasta con las cuerdas fundamentales, que se supone que tienen (normalmente) un tamaño muy inferior al del núcleo atómico, del orden de la longitud de Planck para ser más precisos. Una D1-brana, por tanto, no sería como una cuerda fundamental pues debería tener la longitud del universo (o al menos un tamaño muy grande, siendo un posible candidato para "cuerda cósmcia"). El argumento por el cuál una Dp-brana debe tener esa extensión no se explica en los libros de texto (al menos no lo e visto en los que he leido). Se supone que debe tener esa extensión por argumentos de estabilidad. Otra configuración posible (pero menos probable) para una Dp-brana es la de una superficie cerrada. Por motivos de simetria la configuracion mas plausible para una Dp-brana cerrada sería una de forma esférica (que esta vez si podria tener cualquier tamaño).

Bien, ya casi podemos pasar a analizar el artículo en cuestión, unos breves apuntes más antes de ir con él. Las warped geometries", o "warped universes" , o también "brane worlds" son modelos fenomenológicos, inspirados en teoria de cuerdas, en los que se postula que el universo observable es una 3- brana (o una pila de ellas), y mas probablemente una D3-brana (quien quiera ver más detalles puede ir al post del blog sobre el particular). En todo caso se impone en los modelos fenomenologicos de manera "ad-hoc" que la materia del modelo standard no puede salir de la brana (matematicamente se hace usando una delta de dircac, un tipo especial, y muy conocido, de distribución) y sólo la gravedad puede moverse en una dimensión extra que sin ser de un tamaño cósmico si se supone que es mucho mayor que las dimensiones compactificadas. No obstante incluso la gravedad puede moverse de una manera bastante restringida por esas dimensiones. Ha habido bastante trabajo en crear modelos de cuerdas que se ajusten con diversos grados de precisión a los fenomenológicos. En última instancia hay que hacer notar es que estos "warped universes" tienen muchas predicciones, algunas de las cuales podrian ser observables. Qizás la más famosa sea la posible produccion de agujeros negros (o de gusano, cómo mencione en un post reciente) en el LHC. Pero con todo no ofrecen una explicacioin de cómo se habría llegado a esa configuracion. Aparentemente ese es el tipo de cuestiones que pretenden elucidarse en este paper.

El paper empieza analizando propuestas anteriores sobre el particular. Desde unas basadas en propiedades del "worldsheet" de la cuerda (su superficie de universo, análogo a la línea de universo de una particula en relatividad espacial) orginarias de Curren Vafa a otra, que argumenta que las 3 branas son las únicas que en 10 dimensiones no se intersectan con su anti d Dp-bana oligatoriamente (Las Dp-branas son objetos cargados bajo cierto tipo de campos gauge, una anti Dp-brana tendría carga opuesta. además cómo la carga se corresponde con una orientacion de la brna una anti Dp brana estaría rotada pi grados respecto a una Dp brana). Que las branas intersecten se argumenta que puede dar lugar a un mecanismo de "unwind" (la terminologia proviene de que en su modelo simplificado se asume un universo con compactificacion toroidal, y que las brnas que se pueden desenrollar-unwind- del toro). La idea final sería que las branas terminarían desintegrándose a traves de ese mecanismo de unwind y sólo las que no interesectan, las D3-branas sobrevirian en un universo en el que inicialmente estarían presentes todos los tipos posibles de Dp-Branas. En este punto es importante señalar que el hechode hablar de Dp branas automticamente esta seleccioinado un tipo especial de teoria de cuerdas. En prticular descarta los modelos heteróticos que no admiten Dp-branas. No deja de resultar curioso pués por otro lado al nivel de reproducir las familias observadas de partículas del modelo standard las cuerdas heteroticas siguen siendo los principaes favoritos. En realidad existe una red de dualidades por los cuales se argumenta que todas las teorias de cuerdas son, en el fondo lo mismo. Uno podría cuestionarse entonce porque estudiar un modelo particular. La repuesta obvia es que aunque son encierto modo equivalentes cada teoria de cuerdas correspondería a un cierto regimen posible de comportaamientos, que son los que describe mejor. El universo estaria descrito, a efectos prácticos, por uno de esos modelos (heteroticos, type I, o type II a y b, o alguna teoria M) y las dualidades servirían para estudiarotros aspectos, nome extenderé más al respecto. Obviamente estos modelos de "naturalización" de las 4 dimensiones basados en Dp-branas seguirian sin explicr porque vivimos en ese tipo particular de teoria de cuerdasy no en uno heterotico, pero bueno, seguiría siendo un muy intersante avance, por supuesto.

Lisa y Karch arguyen que el mecanismo de unwinding tienes una serie de problemas y proponen uno diferente. Su idea parte de un modelo cosmológico estandard tipo FRW (Friedman-Robertson-Walker). Ese modelo asume un universo que contiene materia distribuida de manera homogénea e isótropa (y el mismo cumple estas condiciones). Con estos supuestos las ecuaciones tensoriales de Einstein se reducen a unas relativamente sencillas ecuaciones diferenciales. En esas ecuaciones interviene una función que representa las características de la materia., lo que se conoce como ecuación de estado (asociado al tensor energia momento de las ecuaciones de Einstein, obviamente). Bien, este modelo introduce unas ecuaciones de estado que describen un gas de Dp-Branas. Analiza las caracterísitcas de ese gas y llega a la conclusión de que serían las 3 branas las que más contribuirían en esa ecuación de estado, y que por tanto son las que regirían la evolucion del universo.

Cómo he visto que me estoy alargando demasiad y que convendria describir estos aspectos con algun detalle dejo para un segundo post el resto del analisis