SU(2) x SU(2) = SO(4) and the Standard Model

The Yang-Mills SU(N) equation for field strength is Maxwell’s U(1) Abelian field strength law plus a quadratic term which represents net charge transfer and contains the matrix constants for the Lie algebra generators of the group.  It is interesting that the spin orthogonal group in three dimensions of space and one of time, SO(4), corresponds to two linked SU(2) groups, i.e.

SO(4) = SU(2) x SU(2),

rather than just one SU(2) as the Standard Model would suggest, which is U(1) X SU(2) X SU(3).  This is one piece of “evidence” for the model proposed in, where U(1) is simply dark energy (the cosmological repulsion between mass, proved in that paper to accurately predict observed quantum gravity coupling by a Casimir force analogy!), and SU(2) occurs in two versions, one with massless bosons which automatically reduces the SU(2) Yang-Mills equation to Maxwell’s by giving a physical mechanism for the Lie algebra SU(2) charge transfer term to be constrained to a value of zero (any other value makes massless charged gauge bosons acquire infinite magnetic self inductance if they are exchanged in an asymmetric rate that fails to cancel the magnetic field curls).  The other SU(2) is the regular one we observe which has massive gauge bosons, giving the weak force.

Maybe we should say, therefore, that our revision of the Standard Model is

U(1) x SU(2) x SU(2) x SU(3)


U(1) x SO(4) x SU(3).

As explained in, the spin structure of standard quantum mechanics is given by the SU(2) Pauli matrices of quantum mechanics.  Any SU(N) group is simply a subgroup of the unitary matrix U(N), containing specifically those matrices of U(N) with a positive determinant of 1.  This means that SU(2) has 3 Pauli spin matrices.  Similarly, SU(3) is the 8 matrices of U(3) having a determinant of +1.  Now what is interesting is that this SU(2) spinor representation on quantum mechanics also arises with the Weyl spinor, which Pauli dismissed originally in 1929 as being chiral, i.e. permitting violation of parity conservation (left and right spinors having different charge or other properties).  Much to Pauli’s surprise in 1956 it was discovered experimentally from the spin of beta particles emitted by cobalt-60 that parity is not a true universal law (a universal law would be like the 3rd law of thermodynamics, where no exceptions exist).  Rather, parity conservation is at least violated in weak interactions, where only left handed spinors undergo weak interactions.  Parity conservation had to be replaced by the CPT theorem, which states that to get a universally applicable conservation law involving charge, parity and time, which applies to weak interactions, you must simultaneously reverse charge, parity and time for a particle together.  Only this combination of three properties is conserved universally, you can’t merely reverse parity alone and expect the particle to behave the same way!  If you reverse all three values, charge, parity and time, you end up, in effect, with a left handed spinor again (if you started with one, or a right handed spinor if you started with that), but the result is an antiparticle which is moving the opposite way in time as plotted on a Feynman diagram.  In other words, the reversals of charge and time cancel the parity reversal.

But why did Pauli not know that Maxwell in deriving the equations of the electromagnetic force in 1861, modelled magnetic fields as mediated by gauge bosons, implying that charges and field quanta are parity conservation breaking (Weyl type chiral handed) spinors?  We discuss this Maxwell 1861 spinor in, which basically amounts to the fact Maxwell thought that the handed curl of the magnetic field around an electric charge moving in space is a result of the spin of vacuum quanta which mediate the magnetic force.  Charge spin, contrary to naive 1st quantization notions of wavefunction indeterminancy, is not indeterminate but takes a preferred handedness relative to the motion of charge, thus being responsible for preferred handedness of the magnetic field at right angles to the direction of motion of charge (magnetic fields, according to Maxwell, are the conservation of angular momentum when spinning field quanta are exchanged by spinning charges).  Other reasons for SU(2) electromagnetism are provided in, such as the prediction of the electromagnetic field strength coupling.  Instead of the 1956 violation of parity conservation in weak interactions provoking a complete return to Maxwell’s SU(2) theory from 1861, what happened instead was a crude epicycle type “fix” for the theory, in which U(1) continued to be used for electrodynamics despite the fact that the fermion charges of electrodynamics are spin half particles which obey SU(2) spinor matrices, and in which the U(1) pseudo-electrodynamics (hypercharge theory) was eventually (by 1967, due to Glashow, Weinberg and Salam) joined to the SU(2) weak interaction theory by a linkage with an ad hoc mixing scheme in which electric charge is given arbitrarily by the empirical Weinberg-Gell Mann-Nishijima relation

electric charge = SU(2) weak isospin charge + half of U(1) hypercharge

Figure 30 on page 36 of gives an alternative interpretation of the facts, better consistent with reality.

Although as stated above, SO(4) = SU(2) x SU(2), the individual SU(2) symmetries here are related to simple spin orthogonal groups

SO(2) ~ U(1)

SO(3) ~ SU(2)

SO(4) ~ SU(3)

It’s pretty tempting therefore to suggest as we did, that the U(1), SU(2) and SU(3) groups are all spinor relations derived from the basic geometry of spacetime.  In other words, for U(1) Abelian symmetry, particles can spin alone; and for SU(2) they can be paired up with parallel spin axes and each particle in this pair can then either have symmetric or antisymmetric spin.  In other words, both spinning in the same direction (0 degrees difference in spin axis directions) so that their spins add together, doubling the net angular momentum and magnetic dipole moment and creating a bose-einstein condensate or effective boson from two fermions; or alternatively spinning in opposite directions (180 degrees difference in spin axis directions) as in Pauli’s exclusion principle, which cancels out the net magnetic dipole moment.  (Although wishy-washy anti-understanding 1st quantization QM dogma insists that only one indeterminate wavefunction exists for spin direction until measured, in fact the absence of strong magnetic fields from most matter in the universe is continuously “collapsing” that “indeterminate” wavefunction into a determinate state, by telling us that Pauli is right and that spins do generally pair up to cancel intrinsic magnetic moments for most matter!)  Finally, for SU(3), three particles can form a triplet in which the spin axes are all orthogonal to one another (i.e. the spin axis directions for the 3 particles are 90 degrees relative from each other, one lying on each x, y, and z direction, relative of course to one another not any absolute frame).  This is color force.

Technically speaking, of course, there are other possibilities.  Woit’s 2002 arXiv paper 0206135, Quantum field theory and representation theory, conjectures on page 4 that the Standard Model can be understood in the representation theory of “some geometric structure” and on page 51 he gives a specific suggestion that you pick U(2) out of SO(4) expressed as a Spin(2n) Clifford spin algebra where n = 2, and this U(2) subgroup of SO(4) then has a spin representation that has the correct chiral electroweak charges.  In other words, Woit suggests replacing the U(1) x SU(2) arbitrary charge structure with a properly unifying U(2) symmetry picked out from SO(4) space time special orthogonal group.  Woit represents SO(4) by a Spin(4) Clifford algebra element (1/2)(e_i)(e_j) which corresponds to the Lie algebra generator L_(ij)

(1/2)(e_i)(e_j) = L_(ij).

The Woit idea, of getting the chiral electroweak charges by picking out U(2) charges from SO(4), can potentially be combined with the previously mentioned suggestion of SO(4) = SU(2) x SU(2), where one effective SU(2) symmetry is electromagnetism and the other is the weak interaction.

My feeling is that there is no mystery, one day people will accept that the various spin axis combinations needed to avoid or overcome intrinsic magnetic dipole anomalies in nature are the source of the fact that fundamental particles exist in groupings of 1, 2 or 3 particles (leptons, mesons, baryons), and that is also the source of the U(1), SU(2) and SU(3) symmetry groups of interactions, once you look at the problems of magnetic inductance associated with the exchange of field quanta to cause fundamental forces.


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