Geometry and quantum

The simple reason why non-commutative geometry is relevant to understanding the geometry of spacetime is the key role of non-Abelian gauge theories in the Standard Model of elementary particles and weak and strong forces. Gauge theories modify the gravitational symmetry group, i.e. the group of Diff(M) diffeomorphisms of spacetime, into a larger group G that is a semi-direct product with the group of gauge transformations of the second kind. The search for a geometric interpretation of this group as a group of diffeomorphisms of a higher-dimensional variety is the essence of Kaluza-Klein's idea. Non-commutative geometry provides another avenue.