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In this question, Orbicular made the following comment to Feb7 and my own answers;

Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" insight to physics. The reason is that they only have toy models, all of which are unphysical (in the sense that they predict things which differ from real world measurements). Furthermore even the toy models are usually extremely complicated, killing most expectations to get a "real" model (which is not toyish).

First, I want to thank Orbicular for pointing this out, as it is something that I 'kinda' knew, but often forget. The purpose of this question, is to ask for a deeper explanation, either from Orbicular or someone else. In particular

to what degree does Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry influence/assist 'real' models and actual physics related to the physical world?

I don't wish for this question to turn into a debate about whether or not these maths will later be applied in some beautiful stringy-quantum-symmetry theory; I would much rather it be some explanation of the real use of these things. Specifically, I am interested in hearing about the use of Quantum Groups and their representations to Physicists along with some thoughts on the actual usefulness of the results in NC Algebraic Geometry of those articles I posted over here. Another particularly interesting subject I would like to hear about is the usefulness of commutative algebraic geometry in physics.

Some things I have found

Just two references that I have found that at least address these things to some degree are Peter Woit's lecture notes on Representation Theory, and in Shawn Majid's book on Quantum Groups he discusses some definite physical motivation for studying quantum groups.

Thanks!

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Representation theory per se is ubiquitous in physics and chemistry. They usually abusively call it "group theory". See mathoverflow.net/questions/11784/… for starters. –  Steve Huntsman Feb 8 '10 at 20:11
    
I am familiar with the more classical representation theory applications to physics as in particle physics. I should have been more clear that I am referring to representation of quantum groups and the like. –  B. Bischof Feb 8 '10 at 20:14
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@BB: OK, then consider the Yang-Baxter equation en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation –  Steve Huntsman Feb 8 '10 at 20:18
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I am under the impression that the YB equation first arose in physics, and that quantum group were first invented in connection to quantum integrable systems. But I don't know any references, nor any definite facts, hence this is a comment, not an answer. –  Theo Johnson-Freyd Feb 8 '10 at 21:09
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I think this is a great question but wonder why the same people who have perniciously closed other broad foundational questions haven't jumped on this? Based on previous criteria, shouldn't it at least be a community wiki? –  Ian Durham Feb 8 '10 at 21:12

6 Answers 6

Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.

The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.

The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.

The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.

More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.

Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.

Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab. In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)

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thanks so much for this answer. I will need some time to look it over, but the first reading was very interesting . –  B. Bischof Feb 9 '10 at 2:42
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Your comments concerning classical representation theory are true, obviously. Please keep in mind that the question concerned "the use of Quantum Groups and their representations to Physicists", so Wigner's and Gell-Mann's results do not apply. Neither supersymmetry nor string theory have been observed up to this point. On the contrary, new (questionable) hypotheses have been introduced (like superpartners and additional space-time dimensions). These hypotheses introduce new degrees of freedom, making the observability (and falsifiability) at least more difficult! –  Orbicular Feb 9 '10 at 8:10
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For the record, I believe that one of the simplest (if not the simplest) quantum system with supersymmetry is a non-relativistic spin-1/2 particle confined in a 1-d harmonic potential. This system can most likely be realized experimentally by placing an electron in a special magneto-optical trap. –  Igor Khavkine Feb 12 '10 at 9:54
    
I think it depends on what you mean by supersymmetry. If you mean simply that the spectrum arranges itself into representations of a Lie superalgebra, then sure. But then there are (approximate) supersymmetries in nuclear physics already. I am slightly more conservative and insist that for a system to be supersymmetric, the superalgebra ought to be a "spacetime" superalgebra (or in statistical mechanical systems a "euclidean" superalgebra). This is the case for the Josephson junction, where you have a conformal superalgebra acting conformally on the two-dimensional space. –  José Figueroa-O'Farrill Feb 12 '10 at 10:26

The (apparent, supposed) mathematics behind the fractional quantum hall effect involves the TQFT invariants coming from representations of quantum groups at roots of unity.

Edit: some links for further reading:

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I appreciate your answer as it is exactly what I am interested in working on. I would love it if you could possibly expound on your answer a bit. If you don't want to but you know some links, that would also be greatly appreciated. If you don't know any that's fine too, thanks again!!! –  B. Bischof Feb 9 '10 at 5:15
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A reference you may find interestings, even if they don't use noncommutative geometry so much as C*-algebras: Scattering theory of topological insulators and superconductors by I. C. Fulga, F. Hassler, A. R. Akhmerov arxiv.org/abs/1106.6351. –  Terry Loring Mar 31 '12 at 17:55

Connes and Chamseddine have applied NCG to particle physics directly and made predictions for the Higgs mass. (See, e.g., here.) I would say this counts as "actual physics". Whether or not their predictions will survive is another issue.

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I have to point out that the model of Connes and Chamseddine is based on the standard model, i.e. they use it. Now they get a different expected Higgs mass (it is higher I think). Now suppose the LHC does not find the Higgs at the energies predicted by the (classical) standard model. Then the standard model has to be wrong (and a lot of physics redone). Connes and C. would consider that their triumph, even though the "wrong theory" gave the correct answer. A colleague of mine pointed out that Connes probably knows best how terribly far away he is from a physically reasonable theory. –  Orbicular Feb 8 '10 at 20:16
    
Thank you for that link, it sounds interesting and this is the sort of answer I was hoping for. –  B. Bischof Feb 8 '10 at 20:16
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Orbicular: the framework of Connes and Chamseddine only admits some of the Lagrangians that would be acceptable in quantum field theory, and one of these admissible Lagrangians is the standard model. The observation that it is admissible is what physicists mean when they say that they are "using it." I consider this a predictive model (although, as Connes would admit, it is still very far from a useful theory of physics). –  Peter Shor Jul 31 '10 at 13:19
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Orbicular: The prediction of Connes and Chamseddine for the Higgs mass is only valid in the "great desert" scenario where there are no new particles with masses on the same order as the Higgs, and the existence of dark matter casts doubt on this scenario. The standard model does not actually predict a mass for the Higgs. Only some Higgs masses are consistent with the standard model, and Connes and Chamseddine's prediction falls squarely within this range. However, recent experiments seem to indicate that the Higgs mass is considerably lighter than Connes and Chamseddine's predictions. –  Peter Shor Jul 31 '10 at 13:20

The Chern numbers, Spin Chern numbers and so forth in condensed matter physics are very important in understanding topological insulators. There are many ways to compute this invariants, and some of them come straight out of noncommutative geometry.

See "Disordered topological insulators: a non-commutative geometry perspective" by Emil Prodan, in Journal of Physics A: Mathematical and Theoretical, 44(2011), 113001.

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The most straightforward example of application of quantum groups to real physics is integrable spin chains. For instance, XXX-1/2 spin chain has excitations which transform under fundamental rep of $\mathfrak{su}(2)$. Let us now consider the quantum deformation of this symmetry $U_q(\mathfrak{su}(2))$. It turns out that the system you'll get is the anisotropic XXZ spin chain where the deformation parameter $q$ is related to the anisotropy parameter of the spin chain.

You can think of much more examples for different algebras. But I agree, this is a rare occasion that modern math comes into physics life.

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I might be wrong, but as far as I remember, in his ICM paper Drinfeld provides some motivation from physics for quantum groups.

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It is related with the Peter's answer above. The origin of quantum groups are in papers by L.D. Faddeev's school on quantum spin chains. –  Alexander Chervov Apr 1 '12 at 14:42

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