(How) is category theory actually useful in actual physics? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:30:17Z http://mathoverflow.net/feeds/question/34861 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34861/how-is-category-theory-actually-useful-in-actual-physics (How) is category theory actually useful in actual physics? Steve Huntsman 2010-08-07T19:51:46Z 2010-08-08T10:40:37Z <p>An <a href="http://mathoverflow.net/questions/34843/what-is-realistic-mathematics/34851#34851" rel="nofollow">answer</a> to a recent question motivated the following question:</p> <blockquote> <p>(how) is category theory actually useful in actual physics?</p> </blockquote> <p>By "actual physics" I mean to refer to areas where the underlying theoretical principle has solid if not conclusive experimental justification, thus ruling out not only string theory (at least for the moment) but also everything I could notice on <a href="http://ncatlab.org/nlab/show/higher+category+theory+and+physics" rel="nofollow">this nLab page</a> (though it is possible that I missed something).</p> <p>Note that I do not ask (e.g.) whether or not category theory has been used in connection with hypothetical models in physics. I've read Baez' blog from time to time over the decades and have already demonstrated knowledge of the existence of the nLab. I am dimly aware of stuff like (e.g.) the connection between between Hopf algebras and renormalization, but I have yet to encounter something that seems like it has a nontrivial category theoretic-component and cannot be expressed in some other more "traditional" language.</p> <p>Note finally that I am ignorant of category theory beyond the words "morphism" and "functor" and (in my youth) "direct limit". So answers that take this into account are particularly welcome.</p> http://mathoverflow.net/questions/34861/how-is-category-theory-actually-useful-in-actual-physics/34862#34862 Answer by Noah Snyder for (How) is category theory actually useful in actual physics? Noah Snyder 2010-08-07T19:57:39Z 2010-08-07T19:57:39Z <p>Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at <a href="http://stationq.ucsb.edu/" rel="nofollow">Microsoft's Station Q</a>.</p> http://mathoverflow.net/questions/34861/how-is-category-theory-actually-useful-in-actual-physics/34894#34894 Answer by Aaron Bergman for (How) is category theory actually useful in actual physics? Aaron Bergman 2010-08-08T06:18:54Z 2010-08-08T06:18:54Z <p>Categories (and higher categories) seem to be a good way of expressing the locality of the path integral in physics. In particular, it is the idea of gluing of local structures that is important. This line of thought leads to the axiomatization of (parts of) various QFTs, with the most success in topological and conformal field theories. This idea has its origins with Atiyah, Segal, Baez-Dolan, Freed and probably a ton of other people I'm forgetting. Braided fusion categories as in the previous answer are an example of this in three dimensions. Most recently, there's Lurie's classification of TQFTs in all dimensions in terms of $(\infty,n)$ categories.</p> http://mathoverflow.net/questions/34861/how-is-category-theory-actually-useful-in-actual-physics/34912#34912 Answer by Konrad Waldorf for (How) is category theory actually useful in actual physics? Konrad Waldorf 2010-08-08T10:40:37Z 2010-08-08T10:40:37Z <p>Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.</p> <p>See "J. Fuchs, I. Runkel, C. Schweigert: <em>TFT construction of RCFT correlators I-V</em> " for the full story or, for a summary, Schweigert's ICM talk <a href="http://arxiv.org/abs/math.CT/0602079" rel="nofollow">Categorification and correlation functions in conformal field theory</a>.</p>