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".
This post imported from StackExchange MathOverflow at 2016-03-14 22:40 (UTC), posted by SE-user Zurab Silagadze