Ivancevic_Applied-Diff-Geom

304 Applied Differential Geometry: A Modern Introductionone loses the notion of a single preferred quantum state that could beregarded as the ‘vacuum’; and that the concept of ‘particle’ becomes vagueand/or observer-dependent in a gravitational context. In a gravitationalcontext, vacuum and particle are necessarily ill defined or approximateconcepts. It is perhaps regrettable that this important lesson has not beenyet absorbed by many scientists working in fundamental theoretical physics[Rovelli (1997)].3.10.4.4 New Approaches to Quantum GravityNoncommutative GeometryNoncommutative geometry is a research program in mathematics andphysics which has recently received wide attention and raised much excitement.The program is based on the idea that space–time may have anoncommutative structure at the Planck scale. A main driving force of thisprogram is the radical, volcanic and extraordinary sequence of ideas of A.Connes [Connes (1994)]. Connes observes that what we know about thestructure of space–time derives from our knowledge of the fundamental interactions:special relativity derives from a careful analysis of Maxwell theory;Newtonian space–time and general relativity, derived both from a carefulanalysis of the gravitational interaction. Recently, we have learned todescribe weak and strong interactions in terms of the SU(3)×SU(2)×U(1)Standard Model. Connes suggests that the Standard Model might hide informationon the minute structure of space–time as well. By making thehypothesis that the Standard Model symmetries reflect the symmetry of anoncommutative microstructure of space–time, Connes and Lott are ableto construct an exceptionally simple and beautiful version of the StandardModel itself, with the impressive result that the Higgs field appears automatically,as the components of the Yang–Mills connection in the internal‘noncommutative’ direction [Connes and Lott (1990)]. The theory admitsa natural extension in which the space–time metric, or the gravitationalfield, is dynamical, leading to GR [Chamseddine and Connes (1996)].The key idea behind a non-commutative space–time is to use algebrainstead of geometry in order to describe spaces. Consider a topological(Hausdorf) space M. Consider all continuous functions f on M. Theseform an algebra A, because they can be multiplied and summed, and thealgebra is commutative. According to a celebrated result, due to Gel’fand,knowledge of the algebra A is equivalent to knowledge of the space M,