Alain Connes' Dirac Propagator: Discrete noncommutative quotient nonlocal phase difference frequency

One playlist of Alain Connes that I made
2nd Playlist of Alain Connes - his "Music of Spheres" lectures
"In the noncommutative geometry, we don't have to take a square root, we add a square root which is the inverse of the Dirac Operator...The Dirac Operator is a mass and its inverse is a length. You send a wave from A to B and you look at the maximal shift of the wave with the condition that the wave doesn't vibrate to fast because the commutator is equal or less than 1.....The reason also why it can not commute with the coordinates. Imagine the line element [Fermion Propagator] commutes with the coordinates, because it's infinitesimal the coordinates have to take a specific value...It if is located somewhere what it means, if you want to measure something you have to go to this place and come back...Whereas the Dirac Propagator is not located anywhere and the price to pay is exactly that it does not commute with the coordinates."...
"That time is passing as a variable is not the right answer...when you take an observation at the quantum level - for instance the one slit experiment....you send an electron or photon through a tiny slit and after that slit it goes somewhere - there is a target...because it can't be reproduced."
"The fundamental variable is not in the passing of time but is in this noncommutativity.... it is this quantum uncertainty which is the source of the passing of time...the true origin of noncommutative geometry, ...which is the fact that noncommutativity creates time.... If you take purely imaginary time, this will be a switching of order from xy to yx....When you change the state...automorphisms which are never seen, because they are inner, because they are automatically there when you take the noncommutative algebra....
"The message is very very strong. You don't limit a state. If you limit a state, this would be bad. .. It would not just be the algebra that gives you time. ...JUST the noncommutative algebra by itself generates its own time....The philosophical message tells you that time in fact can be a byproduct of noncommutativity. You don't need anything else. And it's related to quantum mechanics and it's related to what we know of Heisenberg evolution...It's supported by very precise physical equations. The philosophical message is much stronger. The philosophical message is much stronger, because of the coexistence of continuous and discrete that we are noncommutative. It's because of the noncommutativity that time emerges out of the group. Instead of trying to write things down in time, as we do normally. In fact we should start, there is a variability that is much more important than the passing of time and that's a variability that is constantly present by quantum mechanics. ...Every second that passes has an incredible amount of new variability which is unpredictable. I claim that this is why the clocks turn..."
Conférence faite à l'Académie Royale de Belgique. : La géométrie non commutative en physique et en théorie des nombres. 1. Introduction à la géométrie non commutative
A point is defined by the 2 x 2 matrices... that have an inner noncommutative transformation...as the noncommutative quotient.
In many places in mathematics we are taking inductive limits defined by quotients. There are billions of examples of inductive limits in which the quotient set is not a good set - there are billions of examples...[the limit is a quotient]
the correct power of 2pi x i (imaginary) the total curvature is an integer. The irrational number disappears.
The Lagrange transform is actually a Fourier transform and you only get one eigenvalue (continuous).
inner endomorphism:
" In noncommutative geometry we have inner endormorphisms for any algebra..."
"When you use the power of noncommutativity you get the continuum and you get the continuum with all functions with Matrix values of the 2-sphere...Because the group is slightly noncommutative this group contains a subgroup of inner automorphisms....automatically, out of the noncommutativity, you get for free, the fact that you have to mix gauge transformations, together with gravity, just for free. I told you almost nothing...It feels like sometimes I feel like talking into the void....It just comes from pure thinking about this geometric problem and understanding that the noncommutativity is a virtue, it's not an inconvenience, it's not a problem. It's something that grants you, automatically, we things that you wouldn't have imagined at the start....The relevant geometry is springing out of the quantum. ...And it's springing out of the quantum...because noncommutative is not a problem, it's what we do al the time when we speak...we just have to think differently."

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. From the mathematical viewpoint the Dirac equation can be explained as an eigenvalue equation where the rest
mass is proportional to the eigenvalue of the momentum operator. The Dirac operator can be interpreted as the square root of the Laplace operator. Let us also note here that all mass particles known today (quarks, leptons and neutrinos) are fermions and hence they are described by the Dirac equation. The inverse of the Dirac operator is called the Dirac propagator in physics (or 2 point function). This is nothing other than the Green function for the (linear) Dirac operator.
The Laplacian operator is called an operator because it does something to the function that follows: namely, it produces or generates the sum of the three second-derivatives of the function.

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Lagrangian mechanics describe the difference between kinetic and potential energies, whereas Hamiltonian mechanics describe the sum of kinetic and potential energies.

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What's fascinating is that every human culture uses the Octave, Perfect Fifth and Perfect fourth music interval as 1:2:3:4 ratios. These ratios are actually noncommutative - something only Fields Medal Math Professor Alain Connes points out. Our exponential math, the inverse of the logarithmic function, originates from the wrong music theory of equal-tempered tuning, the secret origin of irrational geometry. The square root of two was first approximated as the major 2nd or 9/8 cubed as the tritone. This is explained by the Pythagorean Plato book by Ernest McClain. As modern Westernized peoples we fetishize science as universal truth because we think "math works" but the standard symmetric commutative geometry math is what created the ecological crisis. Noncommutativity proves that this math was logically wrong - the nonwestern cultures with their simple nonwestern music actually practice spiritual meditation based on the noncommutativity principles.

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Mathematically speaking Planck's constant measures the noncommutativity between position and
momentum as expressed in Heisenberg's uncertainty principle, thus the connection with

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Viktor T. Toth
IT pro, part-time physicist4y
How could the Big Bang have happened everywhere at once, when according to relativity, simultaneity is frame-dependent?

You are absolutely correct! A coordinate frame in which the Big Bang corresponds to the same specific moment in time everywhere is a coordinate frame that represents a family of comoving observers: Observers who see the microwave background radiation as isotropic. Pick some other coordinate frame and you may very well find that in that coordinate system, the age of the universe is location-dependent. But that is precisely why we don’t usually use such coordinate frames; they are not helpful and they make the calculations unnecessarily complicated.

Now things really do get interesting if the universe is not approximately homogeneous; if there are regions that are significantly over/underdense, then even comoving observers won’t agree on the age of the universe. But that takes us to another topic altogether (inhomogeneous cosmology, void cosmology.)
Noncommutative Minisuperspace Cosmology
Noncommutative Quantum Cosmology
Dirac Operator