The most relevant new of this month was, probably, the announcement of the MAGIC discovery of indications of the possibility that light speed could depend on its frequency. The relevant paper is this.
it has been discussed in some blogs (Lubos, Peter Woit, Sabine) and some forums (for example physics forums or, in Spanish, migui´s forum. See the links of these blog to search them if you are interested). I will not add too much about it, at least not still. Just to state that, if truth, it would, probably, be a signature of a quantum gravity effect, and that would be the first quantum gravity effect ever observed, which is, of course, a very, very important thing.
The reason why these could be a sign of quantum gravity is also an interesting thing. LQG people reach to that conclusion from two sides. The best known one comes from the side of canonical LQG. There they state that the area operator has a discrete spectrum on the kinematical Hilbert space and that is an strong suggestion (but not necessarily a definitive one) that there is a minimum length. They have tried to probe the same result directly for the length operator but until now they have obtained only preliminary results because of redundancies on the quantization procedure (or something like that). Also it has been criticised that in the full, dynamical, Hilbert space the discreteness of the area operator could fail. Of course Lubos Motl and the stringy community knew that these result of discreteness made no sense long time ago. Pity that I have not seen a careful mathematical proof of their wordy arguments. Well, anyway, once you have a minimal length you can make a DSR (double special relativity) theory and get modified dispersion relations for the propagation of particles and you get the desired result presumably observed. DSR theories have many problems and nowadays seem to have derived into something named ESR (extended special relativities) which in some way results into some kind of nonconmutative geometry.
Other way, worst known, in which LQG people arrive to this kind of results is from a certain limit of spin foams which results, agian, into an NCG, see for example these paper for the details.
It is interesting to note that in fact any generic NCG (non conmutative geometry)could, potentially, give rise to a breaking of Lorentz symmetry and, hence, to a frequency dependent speed of light. Critical string theory could result, through a non zero vev (vacuum expected value) of the NS-NS antisymmetric tensor give rise to an effective theory describable by an NCG. Seemingly these would mean that critical string theory could explain these result, but in view that Lubos Motl doesn’t point in these direction maybe I am missing something. Also it is courius to note that critical (super)string theory is formulated in ten dimensions to avoid both, the conformal anomaly and, related to it, the Lorentz invariance. Even thought a solution of it, the NCG limit, violates Lorentz symmetry. Theses is not new, the solutions of a theory need not to respect the symmetries of the lagrangian.
Well, these are ways in which some could achieve the MAGIC ideal result (if the alternative explanations with no new physics could be discarded). But the paper doesn’t rely on any of these. It is related to a very special kind of strings, the Liouville strings. I had a previous knowledge of theses, and also of the fact that the mainstream string community had not in good estimate the works of their mayor proponents, Nanopoulos, Mavramatos, and all. I had inquired some people of the string community about these Liouville strings when it was announced in the CERN courier that they predicted the vacuum frequency dispersion of speed treated in these post (B.T.W. it is important to note that at least in the CERN courier paper the clearly stated that LQG perditions and Liouville string predictions were in the opposite direction, I have not seen people being to precise about these concern now).
At that time, 2003. I preferred to study LQG, which looked very promising at that time, and I didn’t pay mayor attention to Liouvile strings, the treatment of then in the Hatfield book "quantum field theory of point particles and strings" was dissuasory enough. For someone who doesn’t know anything about Liouville strings a quick comment. You treat the conformal factor of the world-sheet metric as an independent field. In the end it results into a Lagrangian for it which resembles something related to something previously known as Liouville field (I don’t know the exact reason). The problem is that the lagrangian contains an exponential of the field so it is very hard to work with it. In the Hatfield book it is shown that these problem is related to the fact that the 2d bosonic fields of the world-sheet don’t decouple from the 2d gravity and that means that you are doing 2D gravity. It was studied through matrix models (not the same matrix models related to M theory) and at least in the epoch of the Hatfield book they didn’t allow a reasonable formulations in D<25, that is, they didn’t allow a formulation of string theory in four dimensions which was the main goal of that theories. Probably later the have had some improvements, still i am not too sure.
My idea, when I first tried to think how Liouville strings could result into the MAGIC effect was very obvious; they brooked Lorentz invariance, which was enough. But I begun to read the papers linked in the announcement article and I got shocked. They talked about decoherence, hawking radiation, the role of the Liouville field as an emergent time and the role of spacetime foam as the source of the effect. Separately any of these statements seem bizarre, but together, well, at last I understood why Lubos didn’t even mention "liouville strings" in his entries about these announcement ;-).
I didn´t read in deep any of the papers of nanopoulos and all, but I have made some partial readings related to some of the aspects. I’ll speak a bit about the spacetime foam (not to confuse with the LQG community spin foams). The idea of spacetime foams dates back to wheeler. The idea is that at small scale the quantum fluctuations of the metric become very important and the plane spacetime disappears. In that "foam" could, at least it is cited so in the usual divulgation, happen unusual things, such as topological changes of spacetime. These could result in the virtual formation, and annihilation, of things such as black holes, and also, wormholes (a bit more on these later).
This is the naive view. Stringy theorists claim that S-duality changes drastically this scenario. The reason is as follows, when you get some string theory in weak coupling you can prove that it´s correspondent in strong coupling is another string theory (or maybe M-theory or F-theory in certain circumstances). For example, the dual object of a fundamental string is a D1-brane. Also there is duality between different branes (including duality between D-branes and NS5 branes). How this duality prevent the spacetime foam? It is somewhat obvious once one thinks about it. The strong coupling theory corresponds to the limit where the spacetime foam would appear and instead of it one gets another string theory. I have not had time to think to much about these, but I see some possible subtleties. The first one is that the S-duality is obtained by some heuristic arguments in perturbative theory which can be extended to non perturbative one with the help of supersymmetry. That raises the question of how much these dualities are related to supersymmetry or to D-branes. These could seem irrelevant, but not necessarily. In open string theories (for example, the bosonic one) D-branes appear as diritlech conditions for the extrems of the open strings. You don´t need supersymmetry and RR charges and all that for the existences of that D-branes. I have no notice of the search of string dualities for the bosonic string, but naively one would spect that still the S-dual of an fundamental bosonic string would, again, be an D-1 brane, but I can´t say for sure.
In fact all these could seem totally uninteresting to someone. I still think that to clarify the exact role of supersymmetry in the dualities is basic, after all pure (non stringy) supersymmetric theories also have branes and maybe there is some kind of S-duality betwen point particles and 0-branes and these would mean that supergravities also forbid the spacetime foam. But by now my concern about spacetime foam will go into anther direction, the (noncritical) Liouville strings.
If the reader makes a google search (s)he will find that in closed string theory there are also D-Branes, introduced as boundary states in the conformal theory (see, for example, my previous post about these theme). The interesting, for this post, fact is that also noncritical (closed bosonic) strings can be shown to have D-branes. If the S-duality depends exclusively in the existence of D-branes that would mean that there is no reason for the existence of spacetime foam in Liouville strings, contrary to the claims of Nanopuollos at all. On the other side if the S-Duality depends on supersymmetry it is expected that (super)Liouville strings (the relevant ones if the theory must reflects the reality which has fermions) would also have S-duality and so no spacetime foam. The obstacle for these would be the existence of D-branes, but as far as I see the Liouvile strings would have the same R-R sector, and charges, and so some kind of D-branes, in the appropriate dimensions, so one would expect definitively no spacetime foam. Well, that is what I expect previous to a careful reading of that papers, why didn´t i did it?
Well, that leads me to the last topic of the post, the wormholes. I assume by now that all my readers have the intuitive idea of what a wormhole is. I´ll make a separate post in this topic anyway. This time I´ll just say a few things. On one side one of the ideas of Wheeler is that they could be formed in the space time foam. After all the continuous evolution of a metric can, in general, drive one for spaces with a certain topology to others with a different one.
But if one studies carefully general relativity one learns that there are some theorems stating that if causality must be respected everywhere there cannot be such transitions in topology. Also quantum tunnelling between different topologies could be ruled out under certain reasonable assumptions. Wheeler seemingly ended by trying to achieve something different. Instead of absolute changes in topology you could search for effective changes, i.e. that the transition point between topologies would be of subplanckian size, but not a point. Anyway, I know that string theory claims that it allows topology changes in space time. And also it describes wormholes. In fact (transversable) wormholes relies for it’s stability in some special matter, or, in the presence of a positive cosmological constant. The discovering of these constant in the actual universe has launched an interest in wormholes in the string community, specially in the Randall-sundrum sceneries. I begun to study some papers, but I got somewhat lost and searched for some guidance. I have found very interesting, and very useful, the book by M. Vissier: "lorentzian wormholes" (1996, springer verlag). It is focused in "relativistics" viewpoint on white holes and I am not sure if he says anything about string theory. Also it is previous (I gues) to the discovery of the accelerating universe. And it is somewhat old. But still so it is being one of the most interesting books I have readed for a time, it is very well writing and results easy to understand, at least if you have a good basic in GR. When I would have readed this book (or a relevant part of it) I will try to read the papers of nanopoulous and comment on them. Oh, yeah, I am also reading occasionally a classical (1993) paper in closed string field theory inthe BV antifield formalism. I have confident knowledge that it is, even today, a relevant paper and that you need to read it if you are interested instring field theory. But compared to the Witten open string field theory, and it´s results about tachyon condensation I find these paper terribly boring, yeah, I know that is of topic to these post but...I needed to say it!!! ;-)
Friday, August 31, 2007
Thursday, August 16, 2007
Maths and physics
Untill now most of the posts inthese blog have been expository. It is time for an expeculative one. I´ll try to give a few musings about the role of math in modern physic.
Before anything else I must say that it is totally ridiculous that while theoretical physics goes into more abstract mathemathics graduate programs of the universities goes towards less level in maths in favour of informatic skills. I don´t meanthat the abbility to program would be not important, but I guess a physics student can learn to program anywhere else and t it totally unnecsary to cope an asignature for that purpose. If somene thinks that the informatic in a physics or mathemathic faculty is anything special all I can say is that I learened to program out of any faculty and when I needed tto pass an exam (in the math faculty) I got a 10 note althought I didn´t go to the classes of the asignature, didn´t study a single minut of it, althoguth when I did the exam I had not programmed a single code line in C (the actual language asked in the asignature) for more than a year. So I am certain that not studing a math asignature to study informatic is a totall mess.
Well, afther these break about informatic let´s go with the actual topic. When I was studiying physic at the faculty I had the good sense to look what people was doing inn research. And the most obvious thing that i noticed was that the level in math was very far beyond that was beeing teached in the faculty. That, together with the fact that I hate the lack of rigour of physics in some aspects lend me to ampliate my math studies. I, firstly, studied some of it by myself (begining by set topology), later I simultaneated studies in the physics and math faculties. In particular I got all the geometry and topology related assignatures. In fact I later beguined to study both geometry and topology to an upper level that the graduate asignatures. Specially I loved diferential and algebraic topology. I also studied somewhat about funtional analisis an related topics (that is, measure theory) but with not too much enthsuaism and only to be able to understand a book, the Galdindo and Pascual two volume exposure of quantum mechanics which uses the full aparatous of hilbert spaces as studied by mathematicians. However I didn´t studied another asignatures which I didn´t find necesary for physics so I didn´t end maths at that time. I prefered, instead, to study things such like nonlinear dynais, which had not place in the curriculum of my faculty. A separate chapter deserves group theory. It was inteh physics faculty curriculum and it was teached along the lines of books such as the hammermesch and similar ones. I just can say that fore someone used to the manifold theory I found so terribly poor the exposition that I had to studi by myself ll the manifold part of the theory, it was a pitty that as cause of it I didn´t get such as good familiarity with things like representations, young tableaux and similars and I needed to relearn it later.
The price to study all these math was that I didn´t get preciselly a brilliant expediente in physicst. I never have got disapointed with that fact because I have found a lot more usefull to learn modern math that having a good and deepd knowledge of such things as electronic, optics, nuclear physics and such that.
But said all these in favour of maths I must point also some negative. The spirit with math and physics at the end of the seventies and the beguining of the eigthies seemed to be that it was neccesary to recover the lost time and that physic could win a lot using the modern math. I totally agree that formulation of general gravity in the old fashioned way, i.e. the tensor calculus of Levy-Civita is a total mess. The language of difernetial manifolds makes a crucial diference at the level of understanding the physical ideas behind general relativity.
Somewhat diferent for me is the formulation of gauge theories as conections in principal fibre bundles. Ok, it is a good tool for actual calculatons of monopole or instanton solutions, but personally I don´t see that it gives too much physical insight, if any, which you couldn´t get in the traditional formulations common to phyisicans. I have not gone too much into the moduli space stuff, beyond that teached in the string theory books, so I can´t say anything about it, butwith that exception I personally don´t find find as fastanstic these formulations as many people seems to think.
Well, in the previous two parrafes I have treated mainly the reformulation of stablished theories in a new language. But that is not all what physics could expect to gain from maths. Afther all the very born of physic such as we know it is intimately related to the born of infinitesimal calculus. The Newton laws simply couldn´t have existed if calculus wouldn´t have beeen created first. A few centuries later there was another physical theory whcih required of a, by the epoch, new area of maths. I speak, of course, of general relativity. Albert Einstein didn´t like when Minkowsky reformulated his special theory of relativity in geometrical terms. But without that reformulation it would have been probably imposible to have reached the formulation of general relativity in terms of tensor calculus, i.e. diferential geometry. That was the second case in history where a radically new branch of math played an important role in physics. But for around tw centuries physic could work with the stairght develoments of the math which had raised it to the existence, calculus. The third case where a new branch of math played an important role in physics was the matrice formulation of quantum mechanics.But said these the truth is that the schröedinger formulations in terms of wavefuntions and diferential equations was mcuh more important. It could be said that hilbert spaces, which for many old fashioned physics is not much more that a combination of linear álgebra and sturn liouville theory, play a crucial role in quantum physics. Well, sure, but still most introductory books in quantum mechanics don´t actually explain what a hilbert space is (for a mathematician taste, I mean).
The next big even where math was crucial for physical developments came from the hand of Murray Gellman and his "eight fold way". The aplication of group theory to make sense of the hadronic zoo had a crucial practical impact. I am no sure of how important group theory was before that. Now most people like to relate the Lorentz group to quantum field theory in an absolutely crucial way. But I guess that in fact It didn´t play a very important role and that the diferential equationsprocedure was mos relevant in the development. I.e. People had the klein gordon equation, and later the Dirac equation, and separatelly the Maxwell equations for electromagnetsim. The fact that they were related to spin 0, 1/2 and 1 representations of the Lorentz group was probably something very secondary. It was not until the introduction of grout theory in the flavour stuff of haronic physicis, and later in yang mills theories that group theory was realised as something important and usefull, but my particular viewpoint is that is importance is somehwat exagerated. In particular I think that grout theory allows some quick calculations that let people to play not as much atention to some aspects of the theories and probably something is lost in the process.
With the rise of string theory modern maths became crucial. At the begining math made a diference. Most physics didn´t know modern math and simply couldn´t follow the results, less to say to participate in the developent. But that times went and now in greater or less extend everybody is familaar with modern maths. These means that wht decides if some can make important contributions to string theory (or other aproachs to quantum gravity) depend more in the usual physical intuition and less in familiarity with abstract math. Althougth I must say that I am very skeptic about how appropiately some of the string theorist have learned modern math. Undoubtly Witten did it (his field medaill proves it) but not everybody is Witten.
I have invoked the name of Witten and I must say some more things about him. Altought some of his works are of aundoubtly physical utility many of them are mostly mathematical. For example topoligcal field theories (which I learened beofre string theory, remember, my faouvourite branch of maths was topology ;-) ) are mainly an aplication of the path integral to a topological problem. Althought TFT are a beatifull theory it´s physical utility somewhat dissapointed to me. But TFT´s are a somewhat special topci, what about the rest of physics? I am begining to belive that people still are in the initial beief that by simply appliying new branchs of math they would automathically get new physics. And it is not working, as somewhat would expect. In these point I clearly disagree with Lubos Motl. Le´ts go with an example. algebraic gemoetry. People didn´t learn algebraic gemetry and sudenlly decided to searchwhere to use it. It worked somewhat in the reveres direction. There was a proble, to make proper sense of compactifications in orbifold gometries and reomve some kind of singularities. And them algebraic geometry, and blowing up of singularities came to the rescue. B.T.W. if am skeptic of how properly physicans have learend some branchs of math, topolgy, difeential geometry, my doubts increase when we go to algebraic geometry (by algebraic geometry I understand it inhis full formalism, varieties in arbitrary fields, scheme theory, and not only the special case of complex diferential geometry, or it´s still most reduced subjecto of Rienman surfaces) . By now my understanding of it comes from what is teached in string theory books and some clarifications that a friend of me, who is doing a thesis in number theory (which requires hughs amounts of algebraic geometry) did to me of some aspects.
Anyway, the thing is that physics have almost exausted all the branchs of mathematics (they are using even some absturse areas such as p-adic numbers in things such as p-adic-or adelic-strings or topological geometrodynamics). I don´t think that looking towards the few remote areas which, maybe, still have not been exploited will make any diference (particularly I seriously doubt that category theory would be something which will give anything relevant to physics). Perhaps the only exception would be some areas of math which are somehwat beyond the usual scope of theoretical physicians, nonlinear systems and complexity (in a broad sense which covers things such as markov/stocasthic process, grahp theory, etc) could become relevant. These rise agian the role of group theory. Group theory is importan when symmetry is the key ingrediente. But in nonlinear sciences symmetry is not such important. Maybe playing more atention to nonlinear sciences could force to an small change of paradigm to theorethical phyisics or maybe not.
A place where certianly complexity should play a role (I am aware that some papers ahve already gone trought these line) is in the landscape problem. I totally agree with the skeptics about the utilitie of the anthropic principle. If there is no way to remove the landscape the apropiate tools to investigate it whould be complexity theories and not any kind of anthropic principle. I have had the luck to teach math to people working in biology and for sure all their lines of reasoning are much more addequat to trate all the landscape questions that that stupid anthropic principle. But I hope that someone would find, and if possilble soon, a diferent solution to the cosmological constant problem that the landscape idea.
But with that possible exception I gues that it is time for physicist triying to actually do physics and not relay on looking into mathemathcians to search for their new "revolution". Afther all there was a short epoch where the game worked in the opposite direction. I am talking about Dirac and it´s extensevely used "Dirac´s funtion" which when formalised by the mathematicians became the distirbution theory and about the Feynman path integrals whose proper formalization suposed a lot of hard work in measure theory. Physicians could use these math in the non formal treatement which mathemathicians developed later with a lot of success. In fact many still do it and don´t care at all about the more sophisticated versions. B.T.W. I mentined before that the fibre theory formulation of gauge theories wasn´t, in my opinion, too relevant for physics. But it has been very usefull for mathemathicians (there are a lot of docotrants doing his thesis about those topics). Also, seemengly, applies with some aspects of stiring theory. But I am not sure that these cases are the same that dirac delta funtion or path integrals. Another important aspect is whether most mathemathicians could understand relatively well classical physics and evenquantum mechanics (and certainly general relativty) but I seriously doubt they understand properly gauge theories, the QFT aspects of it, or string theory so the relation, or relevance of the interplay betwen these theories and maths, from the mathemathicals viewpoint is more obscure.
I have not been as organized in the exposition of the ideas that I wanted to express as I would have liked, but hope it sitll there is some coherence in the post.
Before anything else I must say that it is totally ridiculous that while theoretical physics goes into more abstract mathemathics graduate programs of the universities goes towards less level in maths in favour of informatic skills. I don´t meanthat the abbility to program would be not important, but I guess a physics student can learn to program anywhere else and t it totally unnecsary to cope an asignature for that purpose. If somene thinks that the informatic in a physics or mathemathic faculty is anything special all I can say is that I learened to program out of any faculty and when I needed tto pass an exam (in the math faculty) I got a 10 note althought I didn´t go to the classes of the asignature, didn´t study a single minut of it, althoguth when I did the exam I had not programmed a single code line in C (the actual language asked in the asignature) for more than a year. So I am certain that not studing a math asignature to study informatic is a totall mess.
Well, afther these break about informatic let´s go with the actual topic. When I was studiying physic at the faculty I had the good sense to look what people was doing inn research. And the most obvious thing that i noticed was that the level in math was very far beyond that was beeing teached in the faculty. That, together with the fact that I hate the lack of rigour of physics in some aspects lend me to ampliate my math studies. I, firstly, studied some of it by myself (begining by set topology), later I simultaneated studies in the physics and math faculties. In particular I got all the geometry and topology related assignatures. In fact I later beguined to study both geometry and topology to an upper level that the graduate asignatures. Specially I loved diferential and algebraic topology. I also studied somewhat about funtional analisis an related topics (that is, measure theory) but with not too much enthsuaism and only to be able to understand a book, the Galdindo and Pascual two volume exposure of quantum mechanics which uses the full aparatous of hilbert spaces as studied by mathematicians. However I didn´t studied another asignatures which I didn´t find necesary for physics so I didn´t end maths at that time. I prefered, instead, to study things such like nonlinear dynais, which had not place in the curriculum of my faculty. A separate chapter deserves group theory. It was inteh physics faculty curriculum and it was teached along the lines of books such as the hammermesch and similar ones. I just can say that fore someone used to the manifold theory I found so terribly poor the exposition that I had to studi by myself ll the manifold part of the theory, it was a pitty that as cause of it I didn´t get such as good familiarity with things like representations, young tableaux and similars and I needed to relearn it later.
The price to study all these math was that I didn´t get preciselly a brilliant expediente in physicst. I never have got disapointed with that fact because I have found a lot more usefull to learn modern math that having a good and deepd knowledge of such things as electronic, optics, nuclear physics and such that.
But said all these in favour of maths I must point also some negative. The spirit with math and physics at the end of the seventies and the beguining of the eigthies seemed to be that it was neccesary to recover the lost time and that physic could win a lot using the modern math. I totally agree that formulation of general gravity in the old fashioned way, i.e. the tensor calculus of Levy-Civita is a total mess. The language of difernetial manifolds makes a crucial diference at the level of understanding the physical ideas behind general relativity.
Somewhat diferent for me is the formulation of gauge theories as conections in principal fibre bundles. Ok, it is a good tool for actual calculatons of monopole or instanton solutions, but personally I don´t see that it gives too much physical insight, if any, which you couldn´t get in the traditional formulations common to phyisicans. I have not gone too much into the moduli space stuff, beyond that teached in the string theory books, so I can´t say anything about it, butwith that exception I personally don´t find find as fastanstic these formulations as many people seems to think.
Well, in the previous two parrafes I have treated mainly the reformulation of stablished theories in a new language. But that is not all what physics could expect to gain from maths. Afther all the very born of physic such as we know it is intimately related to the born of infinitesimal calculus. The Newton laws simply couldn´t have existed if calculus wouldn´t have beeen created first. A few centuries later there was another physical theory whcih required of a, by the epoch, new area of maths. I speak, of course, of general relativity. Albert Einstein didn´t like when Minkowsky reformulated his special theory of relativity in geometrical terms. But without that reformulation it would have been probably imposible to have reached the formulation of general relativity in terms of tensor calculus, i.e. diferential geometry. That was the second case in history where a radically new branch of math played an important role in physics. But for around tw centuries physic could work with the stairght develoments of the math which had raised it to the existence, calculus. The third case where a new branch of math played an important role in physics was the matrice formulation of quantum mechanics.But said these the truth is that the schröedinger formulations in terms of wavefuntions and diferential equations was mcuh more important. It could be said that hilbert spaces, which for many old fashioned physics is not much more that a combination of linear álgebra and sturn liouville theory, play a crucial role in quantum physics. Well, sure, but still most introductory books in quantum mechanics don´t actually explain what a hilbert space is (for a mathematician taste, I mean).
The next big even where math was crucial for physical developments came from the hand of Murray Gellman and his "eight fold way". The aplication of group theory to make sense of the hadronic zoo had a crucial practical impact. I am no sure of how important group theory was before that. Now most people like to relate the Lorentz group to quantum field theory in an absolutely crucial way. But I guess that in fact It didn´t play a very important role and that the diferential equationsprocedure was mos relevant in the development. I.e. People had the klein gordon equation, and later the Dirac equation, and separatelly the Maxwell equations for electromagnetsim. The fact that they were related to spin 0, 1/2 and 1 representations of the Lorentz group was probably something very secondary. It was not until the introduction of grout theory in the flavour stuff of haronic physicis, and later in yang mills theories that group theory was realised as something important and usefull, but my particular viewpoint is that is importance is somehwat exagerated. In particular I think that grout theory allows some quick calculations that let people to play not as much atention to some aspects of the theories and probably something is lost in the process.
With the rise of string theory modern maths became crucial. At the begining math made a diference. Most physics didn´t know modern math and simply couldn´t follow the results, less to say to participate in the developent. But that times went and now in greater or less extend everybody is familaar with modern maths. These means that wht decides if some can make important contributions to string theory (or other aproachs to quantum gravity) depend more in the usual physical intuition and less in familiarity with abstract math. Althougth I must say that I am very skeptic about how appropiately some of the string theorist have learned modern math. Undoubtly Witten did it (his field medaill proves it) but not everybody is Witten.
I have invoked the name of Witten and I must say some more things about him. Altought some of his works are of aundoubtly physical utility many of them are mostly mathematical. For example topoligcal field theories (which I learened beofre string theory, remember, my faouvourite branch of maths was topology ;-) ) are mainly an aplication of the path integral to a topological problem. Althought TFT are a beatifull theory it´s physical utility somewhat dissapointed to me. But TFT´s are a somewhat special topci, what about the rest of physics? I am begining to belive that people still are in the initial beief that by simply appliying new branchs of math they would automathically get new physics. And it is not working, as somewhat would expect. In these point I clearly disagree with Lubos Motl. Le´ts go with an example. algebraic gemoetry. People didn´t learn algebraic gemetry and sudenlly decided to searchwhere to use it. It worked somewhat in the reveres direction. There was a proble, to make proper sense of compactifications in orbifold gometries and reomve some kind of singularities. And them algebraic geometry, and blowing up of singularities came to the rescue. B.T.W. if am skeptic of how properly physicans have learend some branchs of math, topolgy, difeential geometry, my doubts increase when we go to algebraic geometry (by algebraic geometry I understand it inhis full formalism, varieties in arbitrary fields, scheme theory, and not only the special case of complex diferential geometry, or it´s still most reduced subjecto of Rienman surfaces) . By now my understanding of it comes from what is teached in string theory books and some clarifications that a friend of me, who is doing a thesis in number theory (which requires hughs amounts of algebraic geometry) did to me of some aspects.
Anyway, the thing is that physics have almost exausted all the branchs of mathematics (they are using even some absturse areas such as p-adic numbers in things such as p-adic-or adelic-strings or topological geometrodynamics). I don´t think that looking towards the few remote areas which, maybe, still have not been exploited will make any diference (particularly I seriously doubt that category theory would be something which will give anything relevant to physics). Perhaps the only exception would be some areas of math which are somehwat beyond the usual scope of theoretical physicians, nonlinear systems and complexity (in a broad sense which covers things such as markov/stocasthic process, grahp theory, etc) could become relevant. These rise agian the role of group theory. Group theory is importan when symmetry is the key ingrediente. But in nonlinear sciences symmetry is not such important. Maybe playing more atention to nonlinear sciences could force to an small change of paradigm to theorethical phyisics or maybe not.
A place where certianly complexity should play a role (I am aware that some papers ahve already gone trought these line) is in the landscape problem. I totally agree with the skeptics about the utilitie of the anthropic principle. If there is no way to remove the landscape the apropiate tools to investigate it whould be complexity theories and not any kind of anthropic principle. I have had the luck to teach math to people working in biology and for sure all their lines of reasoning are much more addequat to trate all the landscape questions that that stupid anthropic principle. But I hope that someone would find, and if possilble soon, a diferent solution to the cosmological constant problem that the landscape idea.
But with that possible exception I gues that it is time for physicist triying to actually do physics and not relay on looking into mathemathcians to search for their new "revolution". Afther all there was a short epoch where the game worked in the opposite direction. I am talking about Dirac and it´s extensevely used "Dirac´s funtion" which when formalised by the mathematicians became the distirbution theory and about the Feynman path integrals whose proper formalization suposed a lot of hard work in measure theory. Physicians could use these math in the non formal treatement which mathemathicians developed later with a lot of success. In fact many still do it and don´t care at all about the more sophisticated versions. B.T.W. I mentined before that the fibre theory formulation of gauge theories wasn´t, in my opinion, too relevant for physics. But it has been very usefull for mathemathicians (there are a lot of docotrants doing his thesis about those topics). Also, seemengly, applies with some aspects of stiring theory. But I am not sure that these cases are the same that dirac delta funtion or path integrals. Another important aspect is whether most mathemathicians could understand relatively well classical physics and evenquantum mechanics (and certainly general relativty) but I seriously doubt they understand properly gauge theories, the QFT aspects of it, or string theory so the relation, or relevance of the interplay betwen these theories and maths, from the mathemathicals viewpoint is more obscure.
I have not been as organized in the exposition of the ideas that I wanted to express as I would have liked, but hope it sitll there is some coherence in the post.
Tuesday, August 07, 2007
The case for D-Branes in closed strings: Boundary States
In a recent post about the "brane forest" I said that although in type II strings, which are closed strings, there are d-branes, and play a very important role, I had not seen an explicit construction of them.
After all in open string theory there is an easy way to see the appearance of d-branes, they are hyperplanes at which the extremes of open string can end. But, clearly, the same image badly could work in closed string theory because closed strings can leave a d-brane (these is the stringy inspiration for the wrapped brane sceneries ala Randall-Sundrum). It has not been totally trivial to find a minimally satisfactory answer (for my taste) of the question. Before going to it I´ll review some heuristic arguments.
The most common one is based in the analysis of d-branes interactions. These can be understood in different ways. One is by the mediation of open strings joining the two d-branes. Considering a one-loop diagram of these strings is equivalent to consider a tree level diagram of a closed string going between these two d-branes. Ultimately these is related to the fact that poles of open strings diagrams of open strings would correspond to closed strings and that, maybe, open strings alone would not be an self-consistent theory and would need a closed string sector.
Well, anyway, these is as far as many articles and books who review d-brane theory go in these picture. They present some other arguments about the existence of d-branes in closed strings, but before commenting on some of them It is time to present the way in which these picture can be achieved in a math formalism which goes beyond wordy arguments. The trick is the use of a an artifact of conformal field theories known as "boundary states" (for the sake of truth I must say that the book on d-branes of Clifford Jonshon actually mentions it, but is far from presenting it in any clear way). I we go to the original literature we can see that the original paper of Polchinsky on the subject http://arxiv.org/pdf/hep-th/9510017 uses these formalism. For people not familiar with conformal field theories, and particularly with the boundary state formalism I leave a few links:
http://wildcard.ph.utexas.edu/~shaji/papers/misc2.pdf (easy to understand intro to boundary states, it requires previous knowledge of CFT)
http://arxiv.org/pdf/hep-th/0011109 (General introduction to CFT´s, including boundary states, it is more formal/rigorous than the previous. As an advise for casual reader to comment that what in the first paper is called "method of images" i these is called "Stocky conditions").
IF someone doesn’t want to read these papers he can get a "quick version" in the original paper of Polchinsky or, for example in http://arxiv.org/PS_cache/hep-th/pdf/9707/9707068v1.pdf (additional papers could be http://arxiv.org/PS_cache/hep-th/pdf/9510/9510161v1.pdf or http://arxiv.org/PS_cache/hep-th/pdf/9510/9510135v2.pdf).
The argument is as follows, you can consider a diagram of closed strings and impose that the "out state" is not given by a closed string but an BRST invariant operator |B> that inserts
a boundary on the world-sheet and enforces on it the appropriate boundary conditions.
I’ll not go further in the details, and refer the reader to the literature cited before, specially these paper. Just to mention that boundary states utility in string theory is not only to introduce D-branes but are useful for many other purposes (for example in the CFT treatment compactifications). In CFT terms a boundary states has a role very related to it’s name, it imposes conditions in the boundary of the world-sheet. Just for completitude I´ll mention that boundary state formalism is a very "natural" way to introduce D-branes in the string field theory version of strings theories. I hope to write some entries in SFT so I´ll give them the details.
Well, these is the most rigorous way to introduce d-branes in closed strings. But in most of the literature it is not made, why? Probably because it is supposed that CFT is reserved to the "advanced" reader, or maybe because supposedly the other ways are in some why "more physical". For example in CFT you can get a more rigorous way to achieve effective actions for certain approximation of situations of string theories. Aa very related example, the Born-Infield action of the d-brane) but the picture of the fields associated to the extremes of the open string dictating the physic of the branes is more appealing.
Well, these are the more rigorous way, but I had said that there were more considerations that support the existence of d-branes in closed string theories. In the original paper of Polchinsky, after the boundary state formalism treatment, he offers an interesting argument by referring to the Hilbert space which I reproduce here:
Periodically identify some of the dimensions in the type II string:
Xu= Xu + 2πR n=p + 1,...,9 (for a Dp-brane)
Now make the spacetime into an orbifold by further imposing
Xu= -Xu
To be precise, combine this with a world-sheet parity transformation to make an
orientifold. This is not a consistent string theory. The orientifold points are sources for the RR
elds (by the analog of the above arguments for D-branes, but
with the boundary replaced by a crosscap *), but in the compact space these
elds have
nowhere to go. One can screen this charge and obtain a consistent compacti
cation
with exactly 16 D-branes oriented. Now take R->0. The result is the
type I string
A few more comment to conclude. An interesting aspect of the boundary state formalism is that you can get a somewhat dual vision of the paper of the d-branes. Instead of considering interactions between them mediated by strings you can think on them as solitonic solutions of string theory and doing perturbative string theory around that solitonic background. For a review (previous to the introduction of d-branes) in string theory the reader could read these article, String solitons.
I, personally, haven’t still readed it, i have a lot of lectures to do, as well as many things to think about. But one thing, the last one of these post, that I wanted to comment is the following. In the viewpoint of a d-brane as an hyperplane where an open string can move semengly is obvious that a d-brane is an infinitely extended object. The full secenarie of Wrapped universes of Randall-Sundrum (or the ekitropic scenario) support these viewpoint. But as is well known d-branes are not supposed to be rigid objects because in gravity there are not such objects, and in fact, guided by the p-brane scenery of supergravity, and also by the theory of "fundamental" branes you can think of a d-branes as a generalization of an string. From these viewpoint it is not clear at all that the brane as an infinitely extended object would be the right one. In fact the 2 and f5 branes of M theroy (M-branes) are supposed to be fundamental objects (despite the fact that it is very difficult to make sense of a theory which gives an interaction picture for them) which in some limits become ordinary fundamental strings. But ok, M-branes are not d-banes, is there a picture to think about d-branees like "small" objects?.
Well, in ordinary QFT there are solitonic solutions to Yang mil theories. They can be interpreted as monopoles. And they are particles with a finite size and all that, certainly not infinitely extended objects. If-d-branes are solitones for an string it would be natural to think of them as small objects of a certain size and not, at least without further reasons to think so) infinitely extended objects. Even if you think in the hyperplane picture you have that they are hyperplanes for a single string, but, why all the strings would be constrained to the same hyperplane?. If you go to the study of d-bane interactions you see that parallel static d-branes don´t interact, but that when they form some angle they do, and, for example, break some supersymmetries. Also moving parallel d-branes interact with a potential proportional to they speed (these configurations of d-branes can, for example, be used to probe distances smaller than the string size). Books, and reviews, explain (or actually prove) these results, but they don´t mention too much about what to think about their implications. I guess that it we must think about it it could be that d-branes related to the many individual strings in the universe, interact to recombine into a single "big" d-brane, or maybe a few parallel ones, but it is conceivable that "small" d-branes with arbitrary size exist. In the literature of string theory you frequently read about d-branes wrapped around torus (or K3 spaces) when considering black hole entropy counting (i hope to write an entry abou these sometime in the future) but it is not so usual to read about actually "shaping" d-branes. Al I have readed so far is in the divulgative book of Leonard Suskind "the cosmic landscape" stating that d-branes in spherical configurations are instable. In a more formal way the d-branes book of Clifford Jonshon in the last part of the chapter in "d-brane geometry I" consider non hyperplanes configurations of d-branes, related to ALE spaces (asintotically locally eculidean), but it a very indirect treatment and it is not easy to follow the details.
Well, as the reader can easilly deduce I am not an expert in string theory (see, for example, the blogs of Lubos Motl or Jackes Distler linked in these blog if you want ton consult the expers) but I hope that maybe reading to someone who has not all the answers can be also interesting because, maybe, the reader could have arrived himself to the same doubts I have, and maybe, I could have answered some of them partially. Also I think it is interesting to discuss somewhat "old" topics and not only the last papers in arxiv (altlthought I don´t discard to discuss some, of course). In any case, fortunately, these last times I have considerable amounts of time to dedicate to the study of strings (and in less extend other approaches to QG) so I hope I will become a more reliable source of information in the near future ;-).
After all in open string theory there is an easy way to see the appearance of d-branes, they are hyperplanes at which the extremes of open string can end. But, clearly, the same image badly could work in closed string theory because closed strings can leave a d-brane (these is the stringy inspiration for the wrapped brane sceneries ala Randall-Sundrum). It has not been totally trivial to find a minimally satisfactory answer (for my taste) of the question. Before going to it I´ll review some heuristic arguments.
The most common one is based in the analysis of d-branes interactions. These can be understood in different ways. One is by the mediation of open strings joining the two d-branes. Considering a one-loop diagram of these strings is equivalent to consider a tree level diagram of a closed string going between these two d-branes. Ultimately these is related to the fact that poles of open strings diagrams of open strings would correspond to closed strings and that, maybe, open strings alone would not be an self-consistent theory and would need a closed string sector.
Well, anyway, these is as far as many articles and books who review d-brane theory go in these picture. They present some other arguments about the existence of d-branes in closed strings, but before commenting on some of them It is time to present the way in which these picture can be achieved in a math formalism which goes beyond wordy arguments. The trick is the use of a an artifact of conformal field theories known as "boundary states" (for the sake of truth I must say that the book on d-branes of Clifford Jonshon actually mentions it, but is far from presenting it in any clear way). I we go to the original literature we can see that the original paper of Polchinsky on the subject http://arxiv.org/pdf/hep-th/9510017 uses these formalism. For people not familiar with conformal field theories, and particularly with the boundary state formalism I leave a few links:
http://wildcard.ph.utexas.edu/~shaji/papers/misc2.pdf (easy to understand intro to boundary states, it requires previous knowledge of CFT)
http://arxiv.org/pdf/hep-th/0011109 (General introduction to CFT´s, including boundary states, it is more formal/rigorous than the previous. As an advise for casual reader to comment that what in the first paper is called "method of images" i these is called "Stocky conditions").
IF someone doesn’t want to read these papers he can get a "quick version" in the original paper of Polchinsky or, for example in http://arxiv.org/PS_cache/hep-th/pdf/9707/9707068v1.pdf (additional papers could be http://arxiv.org/PS_cache/hep-th/pdf/9510/9510161v1.pdf or http://arxiv.org/PS_cache/hep-th/pdf/9510/9510135v2.pdf).
The argument is as follows, you can consider a diagram of closed strings and impose that the "out state" is not given by a closed string but an BRST invariant operator |B> that inserts
a boundary on the world-sheet and enforces on it the appropriate boundary conditions.
I’ll not go further in the details, and refer the reader to the literature cited before, specially these paper. Just to mention that boundary states utility in string theory is not only to introduce D-branes but are useful for many other purposes (for example in the CFT treatment compactifications). In CFT terms a boundary states has a role very related to it’s name, it imposes conditions in the boundary of the world-sheet. Just for completitude I´ll mention that boundary state formalism is a very "natural" way to introduce D-branes in the string field theory version of strings theories. I hope to write some entries in SFT so I´ll give them the details.
Well, these is the most rigorous way to introduce d-branes in closed strings. But in most of the literature it is not made, why? Probably because it is supposed that CFT is reserved to the "advanced" reader, or maybe because supposedly the other ways are in some why "more physical". For example in CFT you can get a more rigorous way to achieve effective actions for certain approximation of situations of string theories. Aa very related example, the Born-Infield action of the d-brane) but the picture of the fields associated to the extremes of the open string dictating the physic of the branes is more appealing.
Well, these are the more rigorous way, but I had said that there were more considerations that support the existence of d-branes in closed string theories. In the original paper of Polchinsky, after the boundary state formalism treatment, he offers an interesting argument by referring to the Hilbert space which I reproduce here:
Periodically identify some of the dimensions in the type II string:
Xu= Xu + 2πR n=p + 1,...,9 (for a Dp-brane)
Now make the spacetime into an orbifold by further imposing
Xu= -Xu
To be precise, combine this with a world-sheet parity transformation to make an
orientifold. This is not a consistent string theory. The orientifold points are sources for the RR
elds (by the analog of the above arguments for D-branes, but
with the boundary replaced by a crosscap *), but in the compact space these
elds have
nowhere to go. One can screen this charge and obtain a consistent compacti
cation
with exactly 16 D-branes oriented. Now take R->0. The result is the
type I string
A few more comment to conclude. An interesting aspect of the boundary state formalism is that you can get a somewhat dual vision of the paper of the d-branes. Instead of considering interactions between them mediated by strings you can think on them as solitonic solutions of string theory and doing perturbative string theory around that solitonic background. For a review (previous to the introduction of d-branes) in string theory the reader could read these article, String solitons.
I, personally, haven’t still readed it, i have a lot of lectures to do, as well as many things to think about. But one thing, the last one of these post, that I wanted to comment is the following. In the viewpoint of a d-brane as an hyperplane where an open string can move semengly is obvious that a d-brane is an infinitely extended object. The full secenarie of Wrapped universes of Randall-Sundrum (or the ekitropic scenario) support these viewpoint. But as is well known d-branes are not supposed to be rigid objects because in gravity there are not such objects, and in fact, guided by the p-brane scenery of supergravity, and also by the theory of "fundamental" branes you can think of a d-branes as a generalization of an string. From these viewpoint it is not clear at all that the brane as an infinitely extended object would be the right one. In fact the 2 and f5 branes of M theroy (M-branes) are supposed to be fundamental objects (despite the fact that it is very difficult to make sense of a theory which gives an interaction picture for them) which in some limits become ordinary fundamental strings. But ok, M-branes are not d-banes, is there a picture to think about d-branees like "small" objects?.
Well, in ordinary QFT there are solitonic solutions to Yang mil theories. They can be interpreted as monopoles. And they are particles with a finite size and all that, certainly not infinitely extended objects. If-d-branes are solitones for an string it would be natural to think of them as small objects of a certain size and not, at least without further reasons to think so) infinitely extended objects. Even if you think in the hyperplane picture you have that they are hyperplanes for a single string, but, why all the strings would be constrained to the same hyperplane?. If you go to the study of d-bane interactions you see that parallel static d-branes don´t interact, but that when they form some angle they do, and, for example, break some supersymmetries. Also moving parallel d-branes interact with a potential proportional to they speed (these configurations of d-branes can, for example, be used to probe distances smaller than the string size). Books, and reviews, explain (or actually prove) these results, but they don´t mention too much about what to think about their implications. I guess that it we must think about it it could be that d-branes related to the many individual strings in the universe, interact to recombine into a single "big" d-brane, or maybe a few parallel ones, but it is conceivable that "small" d-branes with arbitrary size exist. In the literature of string theory you frequently read about d-branes wrapped around torus (or K3 spaces) when considering black hole entropy counting (i hope to write an entry abou these sometime in the future) but it is not so usual to read about actually "shaping" d-branes. Al I have readed so far is in the divulgative book of Leonard Suskind "the cosmic landscape" stating that d-branes in spherical configurations are instable. In a more formal way the d-branes book of Clifford Jonshon in the last part of the chapter in "d-brane geometry I" consider non hyperplanes configurations of d-branes, related to ALE spaces (asintotically locally eculidean), but it a very indirect treatment and it is not easy to follow the details.
Well, as the reader can easilly deduce I am not an expert in string theory (see, for example, the blogs of Lubos Motl or Jackes Distler linked in these blog if you want ton consult the expers) but I hope that maybe reading to someone who has not all the answers can be also interesting because, maybe, the reader could have arrived himself to the same doubts I have, and maybe, I could have answered some of them partially. Also I think it is interesting to discuss somewhat "old" topics and not only the last papers in arxiv (altlthought I don´t discard to discuss some, of course). In any case, fortunately, these last times I have considerable amounts of time to dedicate to the study of strings (and in less extend other approaches to QG) so I hope I will become a more reliable source of information in the near future ;-).
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