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

Quantum gravity effects in the infra-red: a theoretical derivation of the low energy fine structure constant and mass ratios of elementary particles