Applications of Jordan algebras

Question

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.

Answer 1

I would like to elaborate on the link to "associative" problems (such as the Zelmanov's solution of the restricted Burnside problems mentioned by Tom De Medts) - mainly because by saying that Jordan algebras are nonassociative algebras satisfying a strange list of axioms, you are forgetting that some reverse engineering takes place here.

First of all, if $A$ is an associative algebra, considering the new operation $x\circ y=x\circ y+y\circ x$ is not a completely unnatural thing from the algebraic point of view. Moreover, some important classes, like Hermitian matrices, form a subalgebra with respect to $\circ$, but not with respect to the matrix product!

Then, it is natural to try and axiomatise this operation. For instance, we all know that the operation $xy-yx$ defines Lie algebras, so the Jacobi identity holds. Once you play around with elements a little bit, you discover that the operation $\circ$ satisfies the identity $((x\circ x)\circ y)\circ x = (x\circ x)\circ (x\circ y)$, the Jordan identity you see in the literature. (And that's how that identity emerged.)

What however is very different from the case of Lie algebras is that in fact $\circ$ satisfies many other identities that do not follow from the Jordan identity by algebraic manipulations, for example an identity of degree $8$ discovered by Glennie in early 1960s (https://en.wikipedia.org/wiki/Glennie%27s_identity). But it turns out that working within the larger class of algebras satisfying the Jordan identity only is actually not bad, and a lot of things can be accomplished in that more general context.

Answer 2

They turn up quite often in the study of (exceptional) linear algebraic groups. The most famous instance of this is the fact that algebraic groups of type $F_4$ are precisely the automorphism groups of Albert algebras, i.e. $27$-dimensional exceptional Jordan algebras. One possible reference for this fact is the book "Octonions, Jordan Algebras and Exceptional Groups" by Springer and Veldkamp.

I have used Jordan algebras (and also some generalization known as structurable algebras) quite often in my own research, in particular in connection to Moufang sets (these are groups "of rank one" in some sense). For instance, in my paper with Richard Weiss, "Moufang sets and Jordan division algebras", Math. Ann. 335 (2006), no. 2, 415–433, we show (among other things) that every Jordan division algebra $J$ gives rise to a Moufang sets, i.e. there exists a group that you could sensibly denote as $\mathrm{PSL}_2(J)$. (When $J$ is an Albert division algebra, for instance, this group is a linear algebraic group of absolute type $E_7$ of $k$-rank one.)

Another, completely different instance where Jordan algebras played a crucial role, was in Zel'manov's solution to the restricted Burnside problem. See, for instance, here.

Answer 3

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".

Answer 4

Jordan algebras and more generally Jordan pairs and triple systems have applications in the theory of Riemannian symmetric spaces. Some relevant links:

https://en.wikipedia.org/wiki/Hermitian_symmetric_space#Jordan_algebras

https://en.wikipedia.org/wiki/Triple_system#Jordan_triple_systems