Oxford

A program for Geometric Unity is presented to argue that the seemingly baroque features of the standard model of particle physics are in fact inexorable and geometrically natural when generalizations of the Yang-Mills and Dirac theories are unified with one of general relativity.

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra

was introduced as a generalised phase space for a quantum system in the

so-called topos approach to quantum theory. Here, it will be shown that

the spectral presheaf has many features of a spectrum of a

noncommutative operator algebra (and that it can be defined for other

classes of algebras as well). The main idea is that the spectrum of a

nonabelian algebra may not be a set, but a presheaf or sheaf over the

base category of abelian subalgebras. In general, the spectral presheaf

has no points, i.e., no global sections. I will show that there is a

contravariant functor from unital C*-algebras to their spectral

presheaves, and that a C*-algebra is determined up to Jordan

*-isomorphisms by its spectral presheaf in many cases. Moreover, time

evolution of a quantum system can be described in terms of flows on the

spectral presheaf, and commutators show up in a natural way. I will

indicate how combining the Jordan and Lie algebra structures may lead to

a full reconstruction of nonabelian C*- or von Neumann algebra from its

spectral presheaf.