Abstract

The real 15-dimensional Lie group SL(4,R) contains pairwise disjoint subgroups GL(1,R), SU(2), SL(3,R) and SL(2,C). Apart from a technical difference in the real form, which can be adjusted by suitable multiplications by i, these are the same as the groups U(1), SU(2), SU(3) and SL(2,C) that are used in the standard model of particle physics. By removing the naive assumption that disjoint groups necessarily commute, it is possible to reproduce the entire group-theoretical foundation of particle physics inside SL(4,R).

The failure of commutativity provides a mechanism for explaining (some of) the unexplained parameters of the standard model. The representation theory of SL(4,R) provides a classification of particles that is coarser than the standard model, and thereby permits a more flexible view of the relationships between (for example) electron, proton and neutron.

Putting all this together with the local coordinate transformation group of general relativity, which is also isomorphic to SL(4,R), suggests ways in which the two theories may not be as incompatible as has been thought for the past 80 years or so.