This talk will be about two applications of Jordan algebras. The first, to quantum mechanics, follows on from the talk of John Baez. I will explain how time dependence makes use of the associator, and how this is related to the commutator in the standard density matrix formulation.
The associator of a Jordan algebra also determines the curvature of a Riemannian metric on its positive cone, invariant under the symmetry group of the norm (mentioned in the talk of John Baez); the cone is foliated by hypersurfaces of constant norm. This geometry is relevant to a class of N=2 5D supergravity theories (from the early 1980s) which arise (in some cases, at least) from Calabi-Yau compactification of 11D supergravity. The 5D interactions are determined by the structure constants of a euclidean Jordan algebra with cubic norm. The exceptional JA of 3x3
octonionic matrices yields an
exceptional’’ 5D supergravity which yields, on reduction to 4D, anexceptional’’ N=2 supergravity with many similarities to N=8 supergravity, such as a non-compact global E7 symmetry. However, it has a compact `composite’ E6 gauge invariance (in contrast to the SU(8) of N=8 supergravity). An old speculation is that non-perturbative effects break the N=2 supersymmetry and cause the E6 gauge potentials to become the dynamical fields of an E6 GUT. Potentially (albeit improbably) this provides a connection between M-theory, the exceptional Jordan algebra, and the Standard Model.