Mass quantisation from a number operator

The masses of the electron, the up quark, and the down quark, are in the ratio 1 : 4 : 9

This simple fact calls for a theoretical explanation.

A few years back Cohl Furey proved the quantisation of electric charge as a consequence of constructing the states for quarks and leptons from the algebra of the octonions [arXiv:1603.04078 Charge quantisation from a number operator]. The complex octonions are used to construct a Clifford algebra Cl(6) which is then used to make states for one generation of quarks and leptons. The automorphism group G_2 of the octonions has a sub-group SU(3) and these particle states have the correct transformation properties as expected if this SU(3) is SU(3)_color of QCD. Further, (one-third of) a number operator made from the Cl(6) generators has the eigenvalues (0, 1/3, 2/3, 1) [with 0 and 1 for the SU(3) singlets and 1/3, 2/3 for the triplets] allowing this to be identified with electric charge. This proves charge quantisation and the U(1) symmetry of the number operator is identified with U(1)_em. Anti-particle states obtained by complex conjugation of particle states are shown to have electric charge (0, -1/3, -2/3, -1). Thus the algebra describes the electro-colour symmetry for the neutrino, down quark, up quark, electron, and their anti-particles. Note that it could instead be the second fermion generation, or the third generation. Each generation has the same charge ratio (0, 1/3, 2/3, 1).

This same analysis can now be used to show that the square-root of the masses of electron, up and down are in ratio 1:2:3 All we have to do is to identify the eigenvalues of the number operator with the square-root of the mass of an elementary particle, instead of its electric charge. And we also get a classification of matter and anti-matter, after noting that complex conjugation now sends matter to anti-matter. as follows:

Matter, sqrt mass Anti-matter, sqrt{mass}

anti- Neutrino 0 Neutrino 0

Electron 1/3 positron -1/3

Up quark 2/3 anti-up -2/3

Down 1 anti-down -1

Compared to the electric charge case above, the electron and down quark have switched places, and we already have our answer to the mass quantisation question asked at the start of this post. There is again an SU(3) and a U(1) but obviously this is no longer QCD and EM. We identify this symmetry with a newly proposed SU(3)_grav x U(1)_grav whose physical implications remain to be unravelled. [GR is supposed to emerge from SU(2)_R this being an analog of the weak force SU(2)_L].

The group E_6 admits a sub-group structure with two copies each of SU(3), SU(2) and U(1). Therefore, one set is identified with the standard model SU(3) x SU(3) x U(1) [electric charge based] and the other with the newly introduced SU(3)_grav x SU(2)_R x U(1)_grav [sqrt{mass} based]. In the early universe, the separation of matter from anti-matter is the separation of particles with positive square-root mass from particles of negative sqrt mass. This separation effectively converts the vector-interaction of pre-gravitation into an attractive only emergent gravitation.

However, the second and third fermion generations do not have the simple mass ratios (0, 1, 4, 9) unlike the electric charge ratios which are same for all three generations. Why so?! Because mass eigenstates are not the same as charge eigenstates. We make our measurements using eigenstates of electric charge; these have strange mass ratios, eg muon is 206 times heavier than the electron. If we were to make our measurements using eigenstates of square-root mass, we would find that all three generations have the mass ratios (0, 1, 4, 9) whereas this time around the electric charge ratios will be strange. There is a perfect duality between electric charge and square-root mass.

A free electron in flight - is it in a charge eigenstate or a mass eigenstate? Neither! It is in a superposition of both, and collapses to one or the other, depending on what we choose to measure. In fact the free electron in flight does not separately have a mass and a charge; it has a quantum number which could be called charge -sqrt mass, which is the quantum number for the unified force. Unification is broken by measurement: if we measure EM effect then we attribue electric charge to the source. if we measure inertia or gravity, we attribute mass to the source. These statements are independent of energy scale. A classical measuring apparatus emerges from its quantum constituents as a consequence of sufficient entanglement: the emergence of such classical apparatus is the prelude to breaking of unification symmetry. In the early universe, sufficient entanglement is impossible above a certain energy [possibly the EW scale] and it appears as if symmetry breaking depends on energy. This is only an indirect dependence. The true dependence of symmetry breaking is on the amount of entanglement. In our current low energy universe we have both low entanglement systems (quantum, unified) and high entanglement systems (classical, unification broken).