What is the relationship between algebraic geometry and quantum mechanics? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:19:05Z http://mathoverflow.net/feeds/question/4964 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4964/what-is-the-relationship-between-algebraic-geometry-and-quantum-mechanics What is the relationship between algebraic geometry and quantum mechanics? Qiaochu Yuan 2009-11-11T02:20:32Z 2010-01-23T06:09:14Z <p>The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of observables. In what sense is algebraic geometry a "classical" (i.e. commutative) phenomenon? How does intuition from quantum mechanics influence how one should think about algebraic geometry and vice versa? What modern areas of research study their interaction?</p> <p>(This question is loosely inspired by a <a href="http://mathoverflow.net/questions/2557/examples-of-applications-of-the-borel-weil-bott-theorem" rel="nofollow">similar question</a> about representation theory.)</p> http://mathoverflow.net/questions/4964/what-is-the-relationship-between-algebraic-geometry-and-quantum-mechanics/4970#4970 Answer by Ben Webster for What is the relationship between algebraic geometry and quantum mechanics? Ben Webster 2009-11-11T02:57:30Z 2009-11-11T02:57:30Z <p>My understanding is that this is roughly the jumping off point of <a href="http://en.wikipedia.org/wiki/Noncommutative%5Fgeometry" rel="nofollow">noncommutative geometry</a>. There has also been recent work which is more algebraic in spirit than the main body of noncommutative geometry, which tends to be functional analysis. For one entry point, see the paper of Crawley-Boevey, Etingof and Ginzburg on <a href="http://front.math.ucdavis.edu/0502.5301" rel="nofollow">"Noncommutative geometry and quiver algebras."</a></p> http://mathoverflow.net/questions/4964/what-is-the-relationship-between-algebraic-geometry-and-quantum-mechanics/5028#5028 Answer by Urs Schreiber for What is the relationship between algebraic geometry and quantum mechanics? Urs Schreiber 2009-11-11T10:35:33Z 2009-11-11T10:35:33Z <p>I also would say it's non-commutative geometry and in particular the notion of spectral triple that is the geometric formalism directly coming out of quantum mechanics. </p> <p>A sketchy description from this point of view is at <a href="http://ncatlab.org/nlab/show/spectral+triple" rel="nofollow">nLab:spectral triple</a>.</p> http://mathoverflow.net/questions/4964/what-is-the-relationship-between-algebraic-geometry-and-quantum-mechanics/5164#5164 Answer by David Bar Moshe for What is the relationship between algebraic geometry and quantum mechanics? David Bar Moshe 2009-11-12T06:11:51Z 2009-11-12T06:11:51Z <p>There is a very nice paper by Martin Schlichenmaier available at arXiv math/000528, where the author discusses the relationship between quantization on Kaehler manifolds and projective geometry. Basically, a (quantizable) Kaehler manifold M can be embedded into a complex projective space by the Kodaira embedding theorem. The quantum Hilbert space becomes the projective coordinate ring of M. </p>