most recent 30 from http://mathoverflow.net 2013-06-19T11:14:12Z http://mathoverflow.net/feeds/question/14680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor B. Bischof 2010-02-08T19:43:03Z 2012-04-01T12:07:33Z

<p>In <a href="http://mathoverflow.net/questions/14518/applications-of-noncommutative-geometry" rel="nofollow">this question</a>, Orbicular made the following comment to Feb7 and my own answers;</p> <blockquote> <p>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).</p> </blockquote> <p>First, I want to thank Orbicular for pointing this out, as it is something that I 'kinda' knew, but often forget. The purpose of <em>this</em> question, is to ask for a deeper explanation, either from Orbicular or someone else. In particular</p> <blockquote> <p>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?</p> </blockquote> <p>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 <a href="http://mathoverflow.net/questions/10512/" rel="nofollow">here</a>. Another particularly interesting subject I would like to hear about is the usefulness of commutative algebraic geometry in physics.</p> <h2>Some things I have found</h2> <p>Just two references that I have found that at least address these things to some degree are Peter Woit's lecture <a href="http://www.math.columbia.edu/~woit/repthy.html" rel="nofollow">notes</a> on Representation Theory, and in Shawn Majid's book on Quantum Groups he discusses some definite physical motivation for studying quantum groups.</p> <p>Thanks! </p>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/14687#14687 Steve Huntsman 2010-02-08T20:08:14Z 2010-02-08T20:08:14Z

<p>Connes and Chamseddine have applied NCG to particle physics directly and made predictions for the Higgs mass. (See, e.g., <a href="http://arxiv.org/abs/0901.0577" rel="nofollow">here</a>.) I would say this counts as "actual physics". Whether or not their predictions will survive is another issue.</p>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/14713#14713 José Figueroa-O'Farrill 2010-02-08T23:51:12Z 2010-02-08T23:51:12Z

<p>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.</p> <p>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.</p> <p>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.</p> <p>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 <a href="http://dx.doi.org/10.1016/0370-1573%2881%2990092-2" rel="nofollow">Physics Report</a> by Slansky called <em>Group theory for unified model building</em>, which for years was the representation theory bible for particle physicists.</p> <p>More generally, many of the more speculative <a href="http://en.wikipedia.org/wiki/Grand%5FUnified%5FTheory" rel="nofollow">grand unified theories</a> 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.</p> <p>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, <a href="http://arxiv.org/abs/hep-th/0512177" rel="nofollow">this paper</a> and others by various subsets of the same authors.</p> <p>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 <a href="http://en.wikipedia.org/wiki/Josephson%5Fjunction" rel="nofollow">Josephson junction</a> 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!)</p>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/14731#14731 Scott Morrison 2010-02-09T03:08:29Z 2010-02-09T05:58:05Z

<p>The (apparent, supposed) mathematics behind the fractional quantum hall effect involves the TQFT invariants coming from representations of quantum groups at roots of unity.</p> <p><strong>Edit:</strong> some links for further reading:</p> <ul> <li><a href="http://arxiv.org/abs/cond-mat/0610583" rel="nofollow">From String Nets to Nonabelions</a></li> <li><a href="http://dx.doi.org/10.1016/j.aop.2004.01.006" rel="nofollow">A class of P,T-invariant topological phases of interacting electrons </a></li> <li><a href="http://dx.doi.org/10.1090/S0273-0979-02-00964-3" rel="nofollow">Topological quantum computation</a></li> </ul>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/18470#18470 rat 2010-03-17T07:14:08Z 2010-03-17T07:14:08Z

<p>Everyone certainly remembers Einstein's revolution.</p> <p>This was an example of a "real" use of riemannian geometry in physics (probably even before "riemannian geometry" turned into its current shape).</p> <p>Now we are looking/ waiting for another breakthrough, and who knows what kind of math will be used - it may be algebraic geometry, noncommutative geometry, their mix or something else.</p> <p>I believe there are many good expository writings about this.</p>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/54955#54955 Peter 2011-02-09T23:26:32Z 2011-02-09T23:26:32Z

<p>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.</p> <p>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.</p>

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/92766#92766 Terry Loring 2012-03-31T17:39:12Z 2012-03-31T17:39:12Z

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

http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor/92801#92801 Anton Lyubinin 2012-04-01T09:11:20Z 2012-04-01T12:07:33Z

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