Topology vs group theory in physics

I have said in some ocations that I dont´t like too much group theory, and that I prefer to study topology. I am going to try to explain a litle bit about it.

One of them is pureley mathemathical, I prefer to study things like homology, cohomology, sheaves, spectral sequences, cobordism, morse theory, K-theory and mostly all the topology in the world than group theory. Group theory, and it´s representations, is more or less always the same, but there are a lot of diferent topological tools, all of them with their own peculiarities.

But maths is not a reason a physician could accept. Well, let´s go to physical motivations. One of the goodnes of topology is to try to decide what interestign things happen when one goes from the local tothe global. For example infixing a gauge for a gauge theory you have that any gauge fixing is local and that you can´t extend it to the whole of the phase space. One consecuence of it is the so called Gribov ambiguity. In fact there is a problem with this. Most topological theories are intended to study finite dimensional manifolds and the phase space of a gauge theory is not one of them, so the path is not straighforward from the topology books to their applications in this problems. For example, the BRST operator of gauge theories certainly has the same characteristics than the external derivative of diferential forms. You can form the de-Rham cohomology with the external derivative, to prove that it is equivalent to another cohomologies and use it as a calculational tool for finite manifolds. BRST is usefull, certainly, to select physical states, but I don´t see in the literature that it would be applied to analize the topology of the infinite dimensional phase space. Fortunately there are other things where topology, in its more straightforward form, is interesting, relativity.

One of the things that I find more interesting in general relativity is the role of topology change. The Einstein equations fix the metric of space-time, but don´t say anything specific about it´s topology. In most conventional problems the topology is somewhat assumed. For example in ths schwarschild solution you are studiying something which is topologicallly R³, or, if there is a black-hole you could doubt about the nature of the singularity and to deciede to excise it. Schwarschild space tie is static so it is not problematic. But wen you have a non static space-time things are more intringuing. At a given "time" you have an hypersurface (the technical key word is "slicing"). The natural questions is to wonder if time evolutions, given from the Einstein equations, could change the topology of that hypersurfaces from one time to another. That is a very mathemathically interesting problem in classical gravity and it is usually studied using cobordisms betwen the hypersurfaces. Cobordism intuitive idea is very easy to understand, you wonder when two given n-manifolds can be the border of another n+1-manifold. If you don´t whant to learn cobordism you simply can go to the results which states the criteria under a objects more best known to physicans, characterisitic classes. The results are that if you don´t impose aditional conditions you actually can have topology changes. But when you impose things such like causality and similars the things are more obscure and probably topology change is forbiden. I must say that even if you understand the detaills of the math involved it is somewhat surprising to try to thnk into the details. Afther all in one point you have a metric for a manifold, and later a metric fo ra diferent manifold. And somewhere in the midle you must have a metric which would be good for limit cases of that manifolds. Try to imagine a two dimensinal cae. For example the traditional idea of the formation of a whormhole. You beguin with a 2 sphere and you go nearing two points of it (think for simplicity of two opposite antipodal, points. In the limit beofore the change to the new topology you would have an sphere with the two antipodal points idetntifieds. Afther that you would have a torus.

Try to think about what´s going on just in the intermediate betwen the two cases. Inmediately before you have two tangent spaces which in the intermediate point must be identified. How to interpretate that physically? For that urpose I guess that topology is not the better tool and that the answer would rely in algebraic geometry. I am actually triying to learn algebraic geometry (beyond the one exposed in string theory books) so I can´t go further into this.

I have said "whormhole". Certainlly there is a lot of literature about them. But what I have readed doesn´t actually says too much about the topology change. For transversabble (minkowskian) wormholes you simple decide based in natural requirements a guess form for the metric. But it is all static, you describe the formed wormhole, not the process of it formation. Some tiem ago wormholes where considered funny curiosities usefull for science fiction tales. With the observation of the acclerated universe the thing has changed. The reason is that the existence of a cosmological constant, or of phanthom energy (another possible cause of the acelerating universe) give to wormholes a different position in physicis. The reason is that transversable womhholes require the existence of energy violating the weak energy condition. Usually it was thought that alghoutght it was possible to have small amounts of this matter (by means of quantum effects such like the cassimir effect) it was very unlikely to have too much of it. But the acclerated expansion of the universe make svery plausible that this matter exists. For example in the branworld universes you can have five dimensional matter which in five dimesnions doesn´t vilate that condtion but that it seems to do so in four. The main conclusion is that because of many possible realizations womrholes in a acceraated should exist, and that one must try to find a reason to explain why the don´t exist, or, better, to try to search for them. This could be possible, for example, triying to detect "macroscopic monopoles". The reason is that an usula magnetic field in the vecinity of a wormhole could appear as a magnetic monopole for some observers. Semengly there are concrete proposals for searching them, but I am not totally sure about how the are evolving.

By the way, note that I have said that wormholes are being heavilly studied in the braneworld scenaries. This is one of the reasons I have studied more in deep the Horava-Witten models. I didn´t feel confortable by only understandin the phenomemological approach. In the near future I´ll go back to the study of wormholes, both in conventinoal and in braneworld univeres, and not only to it´s static aspects but to try to understand the dynamics of the formation. If you are a fan of science fiction I can give you a reason why the could be interestings. In SF histories they are usually a tool for interestelar travel or a time machine. But if you think about it if you actually tri to construc a wormhole the most probable thing is that you would connect points that are (very) near one of them. Later you could try to separate the throats of the wormhole. They would be usefull as a fast interstellar travel device only afther you carry the throaths that distance by a conventional way (suposing that the inner distance bethwen the throats doesnt grow). But there is a possible, more realistic, and more usfull way to use a wormhole. You could put one throat near the sun surface and the other near the earth. The solar energy going inside the womrhole will not spread so you will send to the earth the same energy of the sun as the area of the throats. A crude estimation shows that a throat of a few metter of radius will send to the earth the energy that the human the whole race needs. Of course you need to use an intermediate energy vector to use that electromagnetic energy, but that are "minor detaills" xD.

Still one could think that wormholes, or topology change in general relativiey is a too restricitive field. I don´t think so. In string theory there is a great problem. Triying to seek a viable compactification (or going into the landscape if you like that things). But if one thinks a bit about it a compactification is ultimatelly a topology change in the underliying space. Almost all of the work is being done to try to seek for good compactifications. You use a lot of topology and algebraic geometry for that. That is because you study the topology of the resulting space, but you dont wonder how you arrive to that compactificated spaces. String theory goes beyond general relativity, but still it could be interesting to see what a general relativity (or something near to it) could say about that transitions. For example, what kind of matter would inforce that compactifications?

Until nw I have mainly mentioned topology, what about group theory? I am not going to alk too much about it, but, instead, I´ll try to relate it somewhat to compactifications. The easiest place where group theory can appear in quantum physics is in the theory of the angular momentum of a nonreltivistic particle. Rotations in space are described by the SO(3) group. Th universal covering group of SO(3) is SU(2). On the other hand when you have a quantum system with rotational invariane it can be said that, just like in the classical counterpart, the angular momentum is conserved. The classical angular momentum is L=rxp. If you go from the classical momentum p to the quantum operator p you have the definition for the quantum angular momentum. Youcan show that their component´s don´t conmute so you can only observe one of them simultaneously. On the other hand the squared angular momentum L² conmutes with all the components so you can observe it. You can connec the physics with the group theory saying that L² is a cassimir operator for a representation, and that you can relate L+- = Lx +- iLy which act as ladder operators, similar to the a+ a- of the harmonic oscilators, to elements of the group theory of SU(2).

This was an external symmetry. Latter, with the advance of particle physics and QFT appeared internal symmetries, and they were related to easy groups. SU(2), SU(3) and U(1). And the first unification attemp was SU(5). The same technicques used for the group theoyr tratement and a few others allowed to do a lot of calculatiions in particle physicis. Another point of group theory was the role played by the poincaré an dLorent group to the classifications of particles. One could relate the spin of a particle to a representations of the Lorentz group. Of course all of this is well known by a lot of the people that wouldmost likely wuld be interested in reading this blog, so which is my point against group theory.

I said that I wuld relate group theory and compactification. Afther all one of the points of stirn gtheory is to recover the ideas of Kaluza-klein. One can show (I did a post, in spanish, in this blog about it) that an five dimensional space compactified in a circle makes that the "external" gravitational of the five dimension becomes an U(1) "internal" simmetyr equivalent to electromagnetism. This is the ideal case. But nature is not allways ideal. In facto it usually isn´t. The hidrogen atom is almost perfectly shperically symmetric, but because of electron, electron-interaction other atoms are not. Still angular momentum is a very usefull quantity, but it doesntt any more is related to a perfect "reall" symmetry.

What about the circle of the compactification?. On could expect that it would be a perfect circle. Afther all symmetric configurations are commonly the right solutions for optimization problems. But if we think in the possible dynamic of the compactification one realize that that circle still i space-time, and because of general relativity space time is dynamic. So one could expect that it will not be an static perfec circle but that it will be subject to fluctuations. But if the guge symmetry really is a reflection of the diffeomorphism invariance of the compactified space and this is subject to fluctuations it can´t be a perfect symmetry. Moreover, group theory is good to describe the state of th symmetry in a given moment. But there is symmetry breaking,and group theory is not good to stdy it´s dynamic. In fact the dynamic of symmetry breaking should be equivalent to the dynamcic of compatification. In fact there are "dictionaries" translating symmetry breaking into D-brane theory language. And there are also ways to see gauge symmetries in terms of branes which go beyond the idea that I explained here. But still I think that the idea aapply. Group theory is ufull to analyze statics. But any internal group theory should be related to a compactifications, and you can use topology to study compactification. And topology is mathematically richer than group theory, so it can let you go further. And the key ingredient of analizying compactifications, or symmetri breakings could be related (or one could nspire in the study of) to the dymanic of topology change in general relativity, and indirectly to whormholes. In fact rolled dimesnions have an asociated casimir effect. Maybe this could be an alternative explanation of the cosmoogical constant, but I don´t see exactly how. By the way, the cassimir effect is anothe rthing topological in characters, as well as the arahano-bhor. And solitons and instantons in gauge theories can be studied by means of topology. Definitivelly I think topology is mre interesting that group theory. And possibly algebraic geometry is necesary to complement topology in some cases.

Afther this post I hope that you will understand why I don´t thnk that Garret-Liisi theoyr could be related to important and deep questions in physics. But of course that is only a tangential point of the post. Ah, the most veterans readers of this blog may answer about "knotted strings". I stil don´t see a clear reason why they couldn´t exist, but I still lack a full understanding of many aspects ofstring theory. Obviously is not the easist thing in the wolrd to understand, but it is interesting, certanly.