In this lecture I put forward two new mathematical approaches to fundamental physics. The first is joint work with G. Moore of Rutgers University and will appear in Surveys in Differential Geometry, volume XII, entitled "A Shifted View of Fundamental Physics".
In this we show how the Dirac operator enables one to introduce retarded and advanced differential equations in a relativistically invariant way. There are two surprising consequencies, one in atomic physics where we find that masses of stable fermions are quantized and the second in cosmology where we seem to encounter the cosmological constant.
The second idea can be seen as a refinement of the Skyrme model of baryons. I propose using 4-dimensional Riemannian manifolds to model matter. Time is ignored and the 4 "spatial" dimensions are those of the Kaluza-Klein 5-dimensional extension of Minkowski space, but with the roles of electricity and magnetism reversed.
The manifolds used are oriented and conformally self-dual, precisely those which have Penrose twistor spaces, and can be treated by complex analysis. Electrically neutral matter (neutrons and neutrinos) is modelled by compact manifolds, while charged matter (protons and electrons) is modelled by complete non-compact manifolds.
More precisely the models with their metrics are:
|
Neutrino |
standard 4-sphere |
|
Neutron |
standard complex projective plane |
|
Electron |
Taub-NUT manifold |
|
Proton |
Atiyah-Hitchin manifold |
All of these have SO(3) symmetry.
Note that the simply-connected Atiyah-Hitchin manifold (which arose as a moduli space of monopoles) is topologically the complement of the real projective plane in the complex projective plane, with its SO(3) symmetry.
The charged models have asymptotic ends in the background Kaluza-Klein 4-space, while the neutral models intersect this background in a surface (2-sphere or real projective plane).
Many aspects including the dynamical evolution and spectral properties remain to be investigated.