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.
Tuesday, July 31, 2007
Tuesday, July 17, 2007
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 Cuvnk form which we can derivate equations of motion of the form:
4αWuv=Tuv.
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.
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 Cuvnk form which we can derivate equations of motion of the form:
4αWuv=Tuv.
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.
Thursday, July 12, 2007
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.
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.
Tuesday, July 03, 2007
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 Cp+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 Bmn. 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 Bmn 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.
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 Cp+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 Bmn. 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 Bmn 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.
Subscribe to:
Posts (Atom)