Because you can an operator with a Discrete Spectrum in Hilbert Space and in the same Hilbert Space an operator with continuous spectrum, but of course they can not commute.
The finite invariant is allowing you to reconstruct the geometry from the spectrum....this is how the notion of point will emerge, by a correlation from different frequencies. The space will be given by the scale, the spectrum. The music of the space will actually be done by the various chords that are possible. It's not enough to give the scale. You also have to give which chords are possible. ...The invariant which is complementary to the spectrum invariant....that's why I call it the scaling invariant. So how do we get back to points in the geometric space.... When you have a point in space you have a matrix of rank four, the matrix of the inner products of that point. A matrix of complex numbers, a nice self-adjoint... A point should be thought of as a correlation between frequencies.....There is a fine structure in spectral....
We are in fact working with a huge microscope (at CERN) ... in order to understand this fine structure...you have to rethink about geometry... geometry is born from Hilbert space and only observable quantities because it chooses invariant quantities in Hilbert Space...
the propagator is the inverse of the Dirac operator... by sending a wave...with a constraint... not faster than one. Less than one. It works for discrete spaces. It works for noncommutative quantum states...
You have a manifold times a finite space.... the finite space is never commutative. ...The simplest finite space which has dimension zero, as far as the spectrum is concerned, because it is finite... 2 x 2 matrix as quaternions plus 4 x 4 matrices [for the inverse frequency eigenvariables] ...
https://sciendo.com/pdf/10.7151/dmgt.2280
Note that the expression is actually abuse of notation for i.e. the matrix inverse of . What you showed is that so you are free to leave out the in your final expression.
https://physics.stackexchange.com/questions/656470/inverse-of-dirac-propagator
the underlying theory behind our approach which uses ideas from quantum field theory and
non-commutative geometry, in particular the notion of an odd K-cycle which is based on the
Dirac operator (and its inverse, the Dirac propagator). Using physics terminology, the key
point in our strategy is this: instead of measuring ordinary length in space-time we measure the
“algebraic (or spectral) length” in the space of quantum states of some fermion acted upon by
the Dirac propagator.
https://arxiv.org/ftp/arxiv/papers/1203/1203.0832.pdf
Connes:
Our brain is an incredible UTENSIL which perceives things in momentum space - all the photons that we receive - and manufactures a mental picture which is geometric. But what I'm telling you, ....I believe from what I said that the fundamental thing is Spectral and that the fundamental thing is of that nature. And that somehow in order to think we have to do this innermost Fourier Transform.... A Fourier Transform not for functions but a Fourier Transform on geometry.... The Music of Shapes is really a Fourier Transform on a Shape and the fact that we have to do it in reverse and this is a function that the brain does amazingly well because we think geometrically.Ritz-Rydberg Law ... Two variables...Some frequencies add....
Heisenberg said ALL frequencies would add as a Group - by Fourier Transform we'd come back to the Hamiltonian of this space that would be a Torus.... Because the Ritz-Rydberg did not allow this then Heisenberg discovered the noncommutative matrices....the quantum observables do not commute.
The Phase Space is a noncommutative system and that's what's behind the space all the time.https://www.noncommutativegeometry.nl/wp-content/uploads/2013/10/ConnesLeiden.pdf
A. Chamseddine and A. Connes, The Spectral action principle, Comm. Math. Phys. 186 (1997), 731–750.
Alain Connes gives Music of Shapes lecture in French
Ritz-Rydberg Law: When expressed in frequencies NOT wavelengths, certain spectral lines add up to give a new spectral line. If you want to understand that kind of law, you had to use not one index (alpha or beta) but TWO indices. If you study spectral lines in that point of view, certain lines are the addition of two different spectral lines. This was a miraculous, a wonderful discovery that was made, thanks to Heisenberg. Heisenberg understood that this law of composition which was called Ritz-Rydberg Law lead immediately to - if you're a physicist you concentrate on observable values - led to Matrix Mechanics. Of course mathematicians know about that but not physicists. If you make a product of two matrices you use precisely this Ritz-Rydberg Law. You obtain the IK from the sum of IJ and JK.
The discovery of Heisenberg was these matrices were not commuting.
The order of the terms...has vital part to play. E=mc(squared) but you can't inverse the terms of this equation in this specific case. Commutativity does no longer hold in the phases of a microscopic system.
This might be a difficult challenge. But we tend to know that kind of phenomenon. Because when we right things down, using language, we know that we have to take into account the order in which we write the letters. If we don't, we have, some times the cases of anagrams.
https://arxiv.org/pdf/1603.00924.pdf
https://core.ac.uk/download/pdf/328855769.pdf
So the inverse of the frequency as noncommutative wavelength or "line element" for the square root that is more dense!! So fascinating!!!
He says it's fairly hard to define the difference between spectrum and "relative spectrum" based on chords....
The universe communicates with us through bar codes ... a very well known example is the red shift - that was sent to these spectra and bar codes that were shifted to the red....
the emergence of spacetime. The Real world started out with mathematics and our very existence is a sort of window into this real world. And this gives us a glimpse of eternity....https://core.ac.uk/download/pdf/328855769.pdf
The spectrum of unbounded self-adjoint operators is similar to
bounded operators, for example, the spectrum of unbounded self-adjoint operators is real.
Unitary operators are bounded linear operators whose spectrum lie on the unit circle. The
spectrum of the unitary operators can then be mapped onto the real line using a Cayley
transform and, hence, the spectrum of an unbounded operator may be written using the
spectral theorem of unitary operators as the basis.
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