  We have recently proposed a pre-quantum, pre-space-time theory as a
matrix-valued Lagrangian dynamics on an octonionic space-time. This theory
offers the prospect of unifying internal symmetries of the standard model with
pre-gravitation. We explain why such a quantum gravitational dynamics is in
principle essential even at energies much smaller than Planck scale. The
dynamics can also predict the values of free parameters of the low energy
standard model: these parameters arising in the Lagrangian are related to the
algebra of the octonions which define the underlying non-commutative space-time
on which the dynamical degrees of freedom evolve. These free parameters are
related to the exceptional Jordan algebra $J_3(8)$ which describes the three
fermion generations. We use the octonionic representation of fermions to
compute the eigenvalues of the characteristic equation of this algebra, and
compare the resulting eigenvalues with known mass ratios for quarks and
leptons. We show that the ratios of the eigenvalues correctly reproduce the
[square root of the] known mass ratios. In conjunction with the trace dynamics
Lagrangian, these eigenvalues also yield a theoretical derivation of the low
energy fine structure constant.
