Lee Smolin recently put up an arXiv article, Matrix universality of gauge and gravitational dynamics, 0803.2926. The first sentence of the abstract is “A simple cubic matrix model is presented,which has truncations that, it is argued, lead at the classical level to a variety of theories of gauge fields and gravity.” A cubic matrix model gives nonlinear field equations. From the Smolin paper:
The dynamics cannot be linear because we want its solutions to reproduce those of non-linear field equations. The simplest non-linear dynamics are quadratic equations, which arise from a cubic action. The simplest possible non-linear action for matrices is
This raises the issue of what happens when one attempts to represent the quantum numbers of the elementary particles as the solutions to quadratic equations. In this post we will show that a very simple set of quadratic equations do give the quantum numbers for the fermions.