Gravity and electromagnetism in noncommutative geometry

Abstract

We present a unified description of gravity and electromagnetism in the framework of a Z 2 noncommutative differential calculus. It can be considered as a "discrete version" of Kaluza-Klein theory, where the fifth continuous dimension is replaced by two discrete points. We derive an action which coincides with the dimensionally reduced one of the ordinary Kaluza-Klein theory.

Chiral spinors and gauge fields in noncommutative curved space-time

“…In our previous work, where we were concerned with only gravity [17,19,18], it was sufficient to have the algebra C ∞ (R, M) C ∞ (R, M). In the present paper, since we are also interested in matter fields coupled to gravity, we choose to begin with the more general algebra (4.1).…”

“…Landi, Viet and Wali [17] have adopted still another approach. It follows closely the standard Riemanian geometry and generalizes it in the case of a two-sheeted space-time that represents a discretized version of Kaluza-Klein theory in which, the fifth compact circular dimension is replaced by two discrete points.…”

An Introduction to Noncommutative Spaces and their Geometries

“…There is no kinetic term for the latter [60]. A somewhat different model of geometry on the Connes-Lott space M ×Y was presented in [71]. The final action is just the Kaluza-Klein action of unified gravity-electromagnetism and consists of the usual gravity term, a kinetic term for a minimally coupled scalar field and an electromagnetic term.…”

“…When computed for a Connes-Lott space M × Y as in [21], the action produces a KaluzaKlein model which contains the usual integral of the scalar curvature of the metric on M, a minimal coupling for the scalar field to such a metric, and a kinetic term for the scalar field. A somewhat different model of geometry on the space M × Y produced an action which is is just the Kaluza-Klein action of unified gravity-electromagnetism consisting of the usual gravity term, a kinetic term for a minimally coupled scalar field and an electromagnetic term [71].…”

Introduction to Non-Commutative Geometry

“…We argued that this was necessary to obtain a satisfactory definition of locality as well as a reality condition. It is possible to relax these requirements and define gravity as a Yang-Mills field [18,120,82,20,24,54] or as a couple of left and right connections [36,37]. If the algebra is commutative (but not an algebra of smooth functions) then to a certain extent all definitions coincide [39,40,81,6].…”