Comment on the dynamics of M38

We survey the possibility of describing a consistent dynamics for the formally real Jordan algebras M3/sup α/, α = 1,2,4,8 of the 3 x 3 Hermitian matrices over the real, complex, quaternionic and octonionic division rings. We particularly examine in detail the dynamics for M38, a quantal system which has been studied so far only from a kinematical point of view. We show that a new kind of dynamics generated by a trilinear form, and with no Hamiltonian, is the only possible description of the most general time evolution. The motion is characterized by a close analogy with the classical motion of four independent harmonic oscillators. We show that difficulties may arise in the definition of time reversal, in both the quaternionic and octonionic case, due to the nature of the underlying division ring. The suggested resolution is to adjoin an imaginary unit commuting with all elements of the Jordan algebra