Grand unification in non-commutative geometry

Abstract

The formalism of non-commutative geometry of A. Connes is used to construct unified models in particle physics. The physical space-time is taken to be a product of a continuous four-manifold by a discrete set of points. The basic algebra is chosen to be an algebra of matrix-valued functions, and the symmetry-breaking mechanism is coded into a generalized Dirac operator. The formalism is then applied to three examples. In the first example the discrete space consists of two points, and the two algebras are taken respectively to be those of 2×2 and 1×1 matrices. With the Dirac operator containing the vacuum breaking SU(2)×U(1) to U(1), the model is shown to correspond to the standard model. In the second example the discrete space has three points, two of the algebras are identical and consist of 5×5 complex matrices, and the third algebra consists of functions. With an appropriate Dirac operator this model is almost identical to the minimal SU(5) model of Georgi and Glashow. The third and final example is the left-right symmetric model SU(2) L×SU(2) R×U(1) B−L.