arXiv subjects
Publications and source records attributed to J. Madore.
A modification of Kaluza-Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a non-trival extension to the total geometry of a linear connection on space-time places severe restrictions on the structure of the noncommutative factor. A counter-example is given.
A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.
The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of nxn complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SUn as an internal space, in a truncated approximation.
The Maxwell vector potential and the Dirac spinor used to describe the classical theory of electrodynamics both have components which are considered to be ordinary smooth functions on space-time. We reformulate electrodynamics by adding an additional structure to the algebra of these functions in the form of the algebra $M_n$ of $n \times n$ complex matrices. This involves a generalization of the notions of geometry to include the geometry of matrices. Some rather general constraints on the reformulation are imposed which can be motivated by considering matrix geometry in the limit of very large $n$. A few of the properties of the resulting models are given for the values $n=2,3$. One of the more interesting is the existence of several distinct stable phases or vacua. The fermions can be quark-like in one and lepton-like in another.