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.
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.