What is the relationship between algebraic geometry and quantum mechanics?

Question

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?

(This question is loosely inspired by a similar question about representation theory.)

Answer 1

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.

A sketchy description from this point of view is at nLab:spectral triple.

Answer 2

My understanding is that this is roughly the jumping off point of noncommutative geometry. 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

• William Crawley-Boevey, Pavel Etingof, Victor Ginzburg, Noncommutative geometry and quiver algebras, Advances in Mathematics 209 Issue 1 (2007) pp 274–336, doi:10.1016/j.aim.2006.05.004, arXiv:math/0502301

Answer 3

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.