Potential applications of modular representation theory to quantum mechanics

Abstract

There is a unique finite group that lies inside the 2-dimensional unitary group but not in the special unitary group, and maps by the symmetric square to an irreducible subgroup of the 3-dimensional real special orthogonal group. In an earlier paper I showed how the representation theory of this group over the real numbers gives rise to much of the structure of the standard model of particle physics, but with a number of added twists. In this theory the group is quantised, but the representations are not. In this paper I consider how a quantisation of the representations might lead to a more fundamental theory.

Comments: 19 pages. v2: 30 pages. Added sections giving details of integral and 3-modular versions of the spinors. Re-organised into sections and subsections for clarity. v3: added a section on experimental evidence. v4: improved notation, added section on possible application to the cosmological lithium problem. v5: added more potential applications. 41 pages