On Discrete noncommutative source of angular momentum: Roger Penrose

 The most obvious physical concept that one has to start with, where
quantum mechanics says something is discrete, and which is connected with
the structure of space-time in a very intimate way, is in angular momentum.

https://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf

Why is linear motion relative but rotation absolute?

This is a deep question that bothered Einstein, and played a role in the inspiration of general relativity. Why, Einstein wondered, is there only one rotational frame that is free of centrifugal force?

Richard Muller

Prof Physics, UC Berkeley, author "Now, The Physics of Time"

He co-wrote the book with Lee Smolin who coined the phrase "fundamental time" that Penrose now uses to explain noncommutativity as protoconsciousness!

Too bad Muller doesn't seem aware of Einstein's other quote:

 Einstein [14] when he wrote

....perhaps the success of the Heisenberg method points to a purely algebraic description of nature, that is, to the elimination of the continuous functions from physics. Then, however, we must give up, in principle, the space-time continuum.

 Foundations of Quantum Theory in the Light of Bohmian Non-commutative Dynamics
B. J. Hiley, 2013

Einstein here attributes the non-commutativity to Heisenberg. The von Neumann-Moyal algebra builds Heisenberg’s non-commutativity into a noncommutative symplectic phase space.
When we consider a non-commutative geometry, we do not have a unique underlying manifold, but we can construct shadow manifolds....

 On the philosophical side, this non-commutative algebra is actually a
mathematical expression of Bohm’s implicate order [3]. The algebra is a
mathematical description of what Bohm calls the implicate order. The
shadow manifolds are examples of what Bohm calls explicate orders arising from the participation of ourselves or our measuring instruments in the
process itself.

 List of participants; Preface; Part I. Introduction: 1. The function of the colloquium - editorial; 2. The conceptual problem of quantum theory from the experimentalist's point of view O. R. Frisch; Part II. Niels Bohr and Complementarity: The Place of the Classical Language: 3. The Copenhagen interpretation C. F. von Weizsäcker; 4. On Bohr's views concerning the quantum theory D. Bohm; Part III. The Measurement Problem: 5. Quantal observation in statistical interpretation H. J. Groenewold; 6. Macroscopic physics, quantum mechanics and quantum theory of measurement G. M. Prosperi; 7. Comment on the Daneri-Loinger-Prosperi quantum theory of measurement Jeffrey Bub; 8. The phenomenology of observation and explanation in quantum theory J. H. M. Whiteman; 9. Measurement theory and complex systems M. A. Garstens; Part IV. New Directions within Quantum Theory: What does the Quantum Theoretical Formalism Really Tell Us?: 10. On the role of hidden variables in the fundamental structure of physics D. Bohm; 11. Beyond what? Discussion: space-time order within existing quantum theory C. W. Kilmister; 12. Definability and measurability in quantum theory Yakir Aharonov and Aage Petersen; 13. The bootstrap idea and the foundations of quantum theory Geoffrey F. Chew; Part V. A Fresh Start?: 14. Angular momentum: an approach to combinatorial space-time Roger Penrose; 15. A note on discreteness, phase space and cohomology theory B. J. Hiley; 16. Cohomology of observations R. H. Atkin; 17. The origin of half-integral spin in a discrete physical space Ted Bastin; Part VI. Philosophical Papers: 18. The unity of physics C. F. von Weizsäcker; 19. A philosophical obstacle to the rise of new theories in microphysics Mario Bunge; 20. The incompleteness of quantum mechanics or the emperor's missing clothes H. R. Post; 21. How does a particle get from A to B?; Ted Bastin; 22. Informational generalization of entropy in physics Jerome Rothstein; 23. Can life explain quantum mechanics? H. H. Pattee; 24. Discussion: phenomena and sense data in quantum theory D. S. Linney and C. F. von Weizsäcker; Index of persons; Index of subjects.

 Back to Penrose:

So we oughtn’t at the outset to have the concept of macroscopic space-
direction built into the theory....

 This example was to some extent stimulated by Bohm’s version of the Einstein-Rosen-Podolsky thought experiment, which it somewhat resembles.) The idea, then, is that any ‘pure
probability’ (if such exists) ought to be something arising ultimately out of a
choice between equally probable alternatives. All ‘pure probabilities’ ought
therefore, to be rational numbers.

To represent all possible directions as states of spin of a spin 1/2 ̄h-bar particle, we need to take complex linear combinations (in the conventional formalism). Here we only use rational numbers—and complex numbers cannot be approximated by
rational numbers alone! Again, the answer seems to be that the space I end
up with is not really the ‘same’ space as the (x, y, z)-space that I could start
with—even though both are Euclidean 3-spaces.