Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ has a uniquely determined closed Jordan ideal $J$ such that $A/J$ (again a JB algebra) can be realized as a subalgebra of the bounded self-adjoint operators on a Hilbert space.
Afterwards I got this awesome question:
Can one model the Schrödinger time-evolution using only JB algebra concepts?
The intuition behind the question is that: Suppose we could represent a JB algebra on a Hilbert space as above (i.e $J$ is trivial). Then, by Stone's theorem, every element of the algebra corresponds to a strongly continuous one-parameter unitary group which in turn corresponds to a continuous automorphism on the JB algebra. The first correspondence is given by the exponential
\begin{equation} e^{ i H t } \end{equation}
for some self-adjoint operator $H$ on the Hilbert space corresponding to an element of the JB algebra and $t \in \mathbb{R}$.
This suggests that (maybe) one could expect to find a canonical mapping between elements of a JB algebra and its continuous one-parameter groups of automorphisms.
Is this true? Note that if $J$ is trivial, it has to be true. Is the nontriviality of $J$ a necessary condition for this to fail? Otherwise what is the proper obstruction?
I tried to google for answers, but I only found remotely related findings.
This question has been studied by Alfsen and Schultz in the form of what they call a dynamical correspondence. It shows that if such a mapping for JB-algebras is suitably well-behaved then the JB-algebra must be isomorphic to the self-adjoint part of a C*-algebra.
A version of it (called observability of energy) has been used in this paper to prove a similar statement but then in finite dimension.
To give some intuition why this fails to hold: if we take the Jordan algebra of symmetric real matrices, then the dimension of the Lie algebra of the automorphisms is smaller than the dimension of the algebra, while if we take the Jordan algebra of self-adjoint quaternionic matrices then the amount of automorphisms is too big. The complex C*-algebras lie just in the sweet spot.