There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral group give
s rise to all the necessary ingredients for a non-relativistic model of quantum mechanics and elementary particles, and how a modification of the binary octahedral group extends this to the ingredients of a relativistic model. Here I investigate the possibility that the binary icosahedral group might be related in a similar way to grand unified theories such as the Georgi--Glashow model, the Pati--Salam model, various $E_8$ models and perhaps even M-theory.
One of the most important advances in our understanding of the physical world arose from the unification of 3-dimensional space with 1-dimensional time into a 4-dimensional spacetime. Many other physical concepts also arise in similar 3+1 relationshi
ps, and attempts have been made to unify some of these also. Examples in particle physics include the three intermediate vector bosons of the weak interaction, and the single photon of electromagnetism. The accepted unification in this case is the Glashow-Weinberg-Salam model of electro-weak interactions, which forms part of the standard model. Another example is the three colours of quarks and one of leptons. In this case, the Pati-Salam model attempts the unification, but is not currently part of the accepted standard model. I investigate these and other instances of 3+1=4 in fundamental physics, to see if a comparison between the successful and unsuccessful unifications can throw some light on why some succeed and others fail. In particular, I suggest that applying the group-theoretical methods of the more successful unifications to the less successful ones could potentially break the logjam in theoretical particle physics.
There are four finite groups that could plausibly play the role of the spin group in a finite or discrete model of quantum mechanics, namely the four double covers of the three rotation groups of the Platonic solids. In an earlier paper I have consid
ered in detail how the smallest of these groups, namely the binary tetrahedral group, of order 24, could give rise to a non-relativistic theory that contains much of the structure of the standard model of particle physics. In this paper I consider how one of the two double covers of the rotation group of the cube might extend this to a relativistic theory.
Finite symmetries abound in particle physics, from the weak doublets and generation triplets to the baryon octet and many others. These are usually studied by starting from a Lie group, and breaking the symmetry by choosing a particular copy of the W
eyl group. I investigate the possibility of instead taking the finite symmetries as fundamental, and building the Lie groups from them by means of a group algebra construction.
