noncommutative time-frequency with matrices: Maurice de Gosson & BG Sidharth on Noncommutative quantum mechanics

 

 

 https://www.sciencedirect.com/science/article/pii/S0022123624001824

 The methods used in this approach were developed by the authors in [2], [3], [12], [13], [17] in the context of a deformation of ordinary quantum mechanics known as noncommutative quantum mechanics.

 

 

 

 https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/apxr.202500099

 the emergence of the deBroglie wavelength. This fol- lows from the noncommutative geometry of space time, rather than the physical Heisenberg Uncertainty Principle.

 B.G. Sidharth

r a spacetime that is discretized or is noncom- mutative. This immediately leads to corrections to the Special Theory of Relativity, more specifically to Lorentz Symmetry....we have deduced from two apparently different view points
namely a noncommutative geometry and a photon mass, are actually the
same. 

 Because of the noncommutative
geometry in (10) space becomes multiply connected, in the sense that
a closed circuit cannot be shrunk to a point within the interval. 

  we started with the Dirac equation and deduced the underlying Noncommutative geometry of spacetime.

 in the usual Quantum Field Theory we encounter divergences which require renormalization. The motivation for the extension of
particles, or the Non commutative structure (4) has been to circumvent these
divergences. In this sense the Non commutative geometry represents ”renormalised” coordinates.

 the generation of mass is due to the noncommutative space-time

 the non-commutative nature of spacetime which manifests itself at ultra
high energies, prohibits a runaway Lorentz Factor

 the space and momentum coordinates become complex indicative of a noncommutative
geometry

 This non commutativity is at the root of the emergence of electromagnetism and spin

 

 Indeed non commutativity arises if there is a minimum space time length as
shown a long time ago by Snyder[11]. What we will argue below is that once
the underlying non commutative nature of the geometry is recognized then
it is possible to reconcile electromagnetism and gravitation.

 https://arxiv.org/pdf/physics/0110040

  It must be observed that the non commutative relations are self evident, in the sense that xy or yx is each of the order of l[squared], and so is their difference because of the non commu-
tativity.
The non commutative or in Witten’s words, Fermionic feature is symptomatic
of the breakdown of the concept of the spacetime points and point particles
at small scales or high energies. As has been noted by Snyder, Witten, and
several other scholars, the divergences encountered in Quantum Field Theory
are symptomatic of precisely such an extrapolation to spacetime points and
which necessitates devices like renormalization. As Witten points out[12],
”in developing relativity, Einstein assumed that the space time coordinates
were Bosonic; Fermions had not yet been discovered!... The structure of
space time is enriched by Fermionic as well as Bosonic coordinates.”
Interestingly it has been shown that the commutative relations (1) lead di-
rectly to the Dirac equation, on the one hand[13]. On the other hand, it
is interesting that a differential calculus over a non commutative algebra
uniquely determines a gravitational field in the commutative limit 

 The effect of the non commutative geometry is therefore to introduce a departure from flat space time,

 non commutativity in momentum components leads to an effect that can be identified with electromagnetism 

 is similar to the torsional formulation of General Relativity[21]. However in this latter case, there is no contribution to the differential interval from the torsional (that is non-commutative) effects. The non-commutative contribution is given by (1) and herein comes
the extended, rather than point like particle.

 BG Sidharth video talk on noncommutative spacetime quantum universe