Wednesday, July 7, 2021

H. Riemann noncommutative de Broglie undertone Pythagorean relativistic ...

"The point of noncommutative geometry is NOT to extend classical concepts and so on, to this situation,...it's more complicated than that. The point is there will be totally NEW phenomenas in the noncommutative case which will have NO commutative counterpart. This is really the important point. And if we look at the hierarchy of the levels of understanding of we have of an ordinary space...I mean - so you see - so something very trivial is the following - when you look at the space you can look at it from various points of view. The closest point of view is Measure Theory, then you have Topology, then you have Differential geometry, by this I mean the soft part, there are currents... Finally you have geometry per se which is the metric aspect....EVEN at the measure theory level, there is a completely NEW phenomenon, which was really the starting point of the whole story, which is the following very striking fact. So I will not talk about spaces - because I will convey all the structure to the corresponding algebra. There is a really striking fact which has probably not passed the mathematical community, which is the following one: IN the Measure Theory - which means it's only a certain kind of algebra which is considered, you are NOT allowed to say what is a continuous function, it's a certain kind of algebra... An algebra A as a God-Given time evolution - so it moves, it rotates, it's a very strange thing. There is an absolutely God-Given morphism from the real line of the Group to the Group OUT of the algebra A which is the Quotient of the automorphisms...and this Group, roughly speaking, what it does is that it exchanges Left and Right... A very striking feature of noncommutative geometry from the Measure Theory point of view...." Fields Medal Alain Connes

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