The characteristic equation of the exceptional Jordan algebra

The attached 1999 paper by Dray and Manogue will perhaps some day be seen as one of the most important papers in theoretical physics. The eigenvalue problem discussed here seems to play a central role in the determination of the coupling constants of the standard model.

The exceptional Jordan algebra (EJA) is the algebra of 3x3 Hermitean matrices with octonionic entries. The algebraic operation is the Jordan product, which is a symmetrized matrix multiplication:

A * B = (AB + BA)/2

The automorphism group of the EJA is the 26 dimensional exceptional group F_4. The automorphism group of the complexified exceptional Jordan algebra is E_6. This same E_6 is also the symmetry group of the Dirac equation for three fermion generations in 10D spacetime (Dray and Manogue, 0911.2255). Hence, the eigenvalue problem for the EJA is also the eigenvalue problem for the Dirac equation in 10D. Its characteristic equation is an elementary algebraic cubic equation whose solutions depend on the trace and the determinant of the 3x3 matrix.

When the octonionic entries in the matrix are the fermionic states and the diagonal entries are value of the electric charge for a fermion, the resulting eigenvalues [all three are real] appear to determine the coupling constants of the standard model [after relating the Dirac equation to the Lagrangian from which it is derived]. Clearly then, the coupling constants are being fixed in 10D, not in 4D. Undoubtedly, quantum systems even at low energies live in 10D. Only classical systems live in 4D.

The eigenvalues take the simple form

q - sqrt{3/8}, q, q + sqrt{3/8}

where q is the value of the electric charge ratio: q is one of (1/3, 2/3, 1) for down quark, up quark, electron.

Reference: Tevian Dray and Corinne Manogue, https://arxiv.org/abs/math-ph/9910004, The Exceptional Jordan Eigenvalue Problem