The group E_6 is the symmetry group of the Dirac equation for three fermion generations, in 10D spacetime, when the states of the fermions are defined in terms of complex octonions. E_6 is also the automorphism group of the complexified exceptional Jordan algebra. The group F_4 is the automorphism group of the exceptional Jordan algebra of 3x3 Hermitean matrices with octonionic entries. Hence the EJA defines the eigenvalue problem for the Dirac equation, and the characteristic equation of the EJA, which is a cubic, determines the eigenvalues. One of these eigenvalues, in conjunction with the Lagrangian, fixes the value of the electric charge, and hence the low energy fine structure constant. Between them, the eigenvalues also determine the mass ratios. This is another piece of evidence that quantum systems do not live in 4D classical spacetime, but in a 10D complex spacetime, at all energies. The coupling constants are fixed in 10D.
Pauli's question to the Devil: `what is the meaning of the fine structure constant?' can be answered as follows. The value of the electric charge of the electron is an eigenvalue of the Dirac equation in ten (complex) spacetime dimensions. More precisely put, the eigenvalue in question is the projection of the `square-root-mass-charge' of the electron, onto the LH U(1)_em sector. The RH sector U(1)_grav fixes mass ratios.
Prior to measurement by a classical apparatus, an electron is in 10D. After measurement, it is in 4D. Collapse of the wave function localises the electron not only in space, but in spacetime, and furthermore, it reduces the dimensionality of the occupied spacetime from 10 to 4, with the penetration depth into the extra dimensions becoming less than a Planck length. Clearly, the collapse of the wave function is a real physical phenomenon, which cannot be described by our current formulation of quantum theory. Any claims to the contrary will have to give a derivation of the values of the coupling constants from conventional QFT. One cannot get away by just saying that these are fixed at very high energies, or that there are multiverses and values of constants are not fundamental!
