Noncommutative geometry and particle physics [Book Review]

The use of noncommutative geometry (NCG) as a tool for constructing particle physics models originated in the 1990s [9, 11]. The main idea can be heuristically regarded as similar to the idea of “extra dimensions” in String Theory, except for the fact that the nature and scope of these extra dimensions is quite different. In the NCG model one considers an “almost commutative geometry”, which is a product (or locally a product in a more refined and more recent version [4]) of a four-dimensional spacetime manifold and a space of inner degrees of freedom, which is a “finite” noncommutative space, whose ring of functions is a sum of matrix algebras. According to the choice of this finite geometry, one obtains different possible particle contents for the resulting physics model. The physical content is expressed through an action functional, the spectral action [5], which is defined for more general noncommutative spaces, in terms of the spectrum of a Dirac operator