TITLE: Applications of Jordan algebras
QUESTION [15 upvotes]: Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\circ B=\frac{1}{2}(AB+BA)$. According to the above article in wikipedia P. Jordan introduced this notion in 1933 to  formalize the notion of an algebra of observables in quantum mechanics.
Question 1. Did this operation become really useful in quantum mechanics in any non-trivial way? 
Question 2. I would be happy to see some explanations why the notion of Jordan algebra is useful/ natural. Are there applications of it to other parts of mathematics?
Here are very few interesting facts I was able to find so far. 
1) There is a really beautiful classification of the finite dimensional formally real Jordan algebras due to  Jordan, von Neumann & Wigner (1934). 
2) The above wikipedia paper mentions a classification of some special class of infinite dimensional Jordan algebras (Zelmanov, 1979). 
3) I have recently heard about the Koecher-Vinberg classification of symmetric cones: such cones are precisely cones of squares in Euclidean Jordan algebras.
Thus the only way I heard Jordan algebras are related to other parts of mathematics is via the classification lists of various subclasses of them.

REPLY [5 votes]: Jordan algebras were originally introduced by Pascual Jordan in a hope to generalize the orthodox formulation of quantum mechanics, but this program was not successful as far as the generalization of quantum mechanics is concerned. In this respect, Jordan algebras are neither essential nor terribly useful in understanding quantum mechanics. 
On the other hand the importance of Jordan algebras in mathematical physics is undisputed and related to the fact that they are tightly interconnected with another nonassociative structures: Lie algebras. As remarked by Kevin
McCrimmon, "if you open up a Lie algebra and look inside, 9 times out of 10 there is a Jordan algebra (of pair) which makes it work”.
In addition to McCrimmon's book "A taste of Jordan algebras", mentioned in comments, the following works might be useful in understanding the role of Jordan algebras in mathematical physics and mathematics:
http://projecteuclid.org/euclid.bams/1183540925 (Jordan algebras and their applications, by Kevin Mccrimmon).
http://arxiv.org/abs/1106.4415 (Jordan structures in mathematics and physics, by Radu Iordanescu).
http://arxiv.org/abs/0809.4685 (Black Holes, Qubits and Octonions, by L. Borsten, D. Dahanayake, M.J. Duff, H. Ebrahim and W. Rubens).
http://www.worldscientific.com/worldscibooks/10.1142/3282 (On the Role of Division, Jordan and Related Algebras in Particle Physics, by F. Gursey and C.-H. Tze). 
The biography of Pascual Jordan can be found here: http://arxiv.org/abs/hep-th/0303241 (Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics, by Bert Schroer). Jordan was a very important figure in developing quantum mechanics and quantum field theory but his reputation was greatly damaged by his relations to the Nazi regime, although, according to Schroer, "he never received benefits for his pro-NS convictions and the sympathy remained one-sided. Unlike the
mathematician Teichmueller, whose rabid anti-semitism led to the emptying of the Gottingen mathematics department, Jordan inflicted the damage mainly on himself".