https://www.youtube.com/watch?v=CPff7CGnjnQ
So the quantum diverges as a discrete differential (Harmonic Series) but because it is noncommutative nonlocality it contains within it a classical integral as a Newtonian continuous limit variable that filters out the line element, now as a wavelength of spacetime. There is no h-bar goes to zero anymore for the classical limit. That continuous limit invariance is then the speed of light of relativity as per Connes....
https://van.physics.illinois.edu/ask/listing/1238
Light travels at the speed of light in vacuum, which is a constant. So there's no speeding up or slowing down....Light will react to a gravitational field and change its direction (again, not really "accelerating" -- Einstein tells us this effect comes from the fact that space and time are not "flat" and the light rays just follow the shortest distance between two points, which may be curved).
So that is how the calculus continuous limit is defined as Steve Strogatz explains - a change of direction without any movement since the length from the center stays the same.
the reality that underlies space must be discrete
Riemann
Georg Friedrich Bernhard Riemann was a German mathematician
the notion of a light ray and the notion of a solid body will not make sense when you look at the very small. You replace a geodesic length by a spectral definition, in the transition quantum frequencies - the corroborate my criticism of Steve Strogatz!
You replace the measurement between two points...by sending a wave from A to B, but by bounding the frequency of this wave using the Dirac Operator...given by the Clifford multiplication....You no longer have to assume that your space is past connected!
You can take the finite difference between the upper layer and lower layer of two dimensional space...
Quantum Two Sphere as 720 degrees
Two Words Listen and Silent are commutative because you are just given a bundle of letters like in Scrabble. So they are converted from noncommutative to commutative just as in noncommutative geometry but it is a tremendously larger world in noncommutative geometry because you differentiate the MEANING of silent and listen.
The quantum two-sphere is much easier to describe in free algebra as 2 x 2 matrices with the square root of negative 1 then to describe the two-sphere as geometry.
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