Introduction to Non-Commutative Geometry

Abstract

A review is made of some recent results in noncommutative geometry, especially those aspects which might in some way be of use in the study of strings and membranes. Efforts to add a gravitational field to noncommutative models of space-time are also reviewed. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a 'shadow' in the commutative limit on the other.

The mystery of the Kerr–Newman metric

“…It also means that the naked singularity is shielded by the fuzzy space time (Dirac's original averages over the zitterbewegung interval) or equivalently the non-commutative geometry (5) (cf. also [15]). …”

Evolution Of The Concept Of Dimension

“…Kaluza-Klein path rather focuses on spatial extension but GR space-time field imply 'field' is fundamental. Madore follows the idea that vector field "can be defined as derivation of the algebra C(V), linear map of C(V) into itself which satisfies the Leibniz rule [7]" Hence space-time is derivation -Heraclites' change -co-necessary to algebraPlato's form -as function/operator, functor/operation dual. Aspect concludes from his EPR observations: "we must get rid of the vision called 'local realist'" [8].…”

The Origin of Mass, Spin and Interaction

“…In the latter approach there is a minimum spacetime cut off, which, as shown nearly sixty years ago by Snyder leads to what is nowadays called a non commutative geometry, a feature shared by the Fuzzy Spacetime also [1,2,3,4,5,6]. The new geometry is given by…”