Basil J. Hiley responds 50 minutes in
The [noncommutative] unfolding is the INvariant behind the principle of coordinates...
An extended [noncommutative] structure...a nonlocal extended energy....
the blob and the void - things don't have to be precise....
the classical world would mean a dead world... with no movement!! Zeno's criticism was correct...
Fritz London, 1927. Dirac got it from there and David Bohm got it from Dirac....
The derivative is no longer continuous....
The Gromov No-Squeezing Theorem.... under classical dynamics (Hamiltonians) - you can't change the "blob" of position and momentum's area.... so there is already an uncertainty that does not allow a point because when h-bar goes to zero there is infinity divergence - "the whole thing blows up."
We are not following the mathematics in a dynamical sense - the process is in the [noncommutative] algebra - what is this algebra telling us in the metaphysical sense....
Fritz London's paper translated from
London, F. Quantenmechanische Deutung der Theorie von Weyl. Z. Physik 42, 375–389 (1927)
London, F. Quantenmechanische Deutung der Theorie von Weyl. Z. Physik 42, 375–389 (1927). https://doi.org/10.1007/BF01397316
additively-periodic with a whole-number multiple of the Planck constant as its period.
When one gets serious about the radical continuum conception of matter, with the solution of the discontinuously-bounded electron in field quantities that vary continuously in space and time, as was suggested by de Broglie’s theory and more consistently by the theory that Schrödinger considered later (1), one will arrive at an especially-definitive complication when one examines the sense that one might assign to metric statements inside of the undulatory continuum, if at all.
That is because in that oscillating and fluctuating infinitely-broad medium that enters in place of the bounded electron, one finds no discontinuities that cannot be understood nor rigid bodies that might permit one to establish a metric as a reproducible yardstick.
It is known to lead to compelling reasons for reinterpreting the entire undulatory formalism statistically, which was proposed by, above all, Born and his collaborators. To the extent that the charge density can be reinterpreted as a statistical weighting function, it is not difficult to see that this indeterminacy in regard to the applicability of the law of identity to which we will refer here must be translated accordingly. However, since that conception initially rejects any interpretation in space and time, there is little of interest in its relation to Weyl’s theory of space.
The law of identity is not applicable to the πάντα ῥεῖ (†) of standing and travelling waves, since there are no features of a continuum that would be suitable for defining a reproducible measurement.
However, I would not like to pause to discuss here whatever it should mean that every line segment is regarded as a complex quantity and what it should mean that the entire Weyl variability of the measure of the line segment is presented as a change in only the phase while preserving the absolute value.
One can now foresee already how that difficulty must be resolved: Quantum theory allows matter to have only a discrete series of equations of motion, and one suspects that those distinguished motions will allow one transport the gauge only in such a way that the phase will have made precisely a whole number of circuits upon returning to the starting point, such that despite the non-integrability of the transport of line segments, the gauge will always be realized in a single-valued way at every location.
despite the non-integrability of the transport of line segments, the gauge
will always be realized in a single-valued way at every location. In fact, one recalls the resonance property of the de Broglie waves, which is the same way that the older Sommerfeld-Enstein quantum condition first reinterpreted the de Broglie condition so successfully. That is generally coupled with the phase velocity, but as a result of the five-dimensional extension of the wave function, the oscillation process will be dispersion-less, and as a result, our current velocity will be identical to the phase velocity.
OH OK so the source of Dirac's Standard Ket was Fritz London splitting the Schrodinger complex wave function into the "real" and "imaginary" parts just as Bohm did to discover the quantum nonlocal potential!! wow.
https://www.academia.edu/111236791/Structure_Process_Weak_Values_and_Local_Momentum
So Hiley cites Fritz London in this 2016 paper....
Conclusions
The concept of a local momentum has been around for many years, but it seems to have been ignored and dismissed as not being meaningful in a quantum context. Indeed Fritz London [34] as long ago as 1945 wrote:The local mean velocity has no true quantum mechanical significance since it cannot be expressed as the expectation value of any linear operator.In other words it has been excluded on the grounds that it does not fit into standard quantum mechanics. In this paper we have shown that the standard quantum formalism is a fragment of a larger non-commutative algebraic structure.
OK but he's quoting 1945 Fritz London yet in that 2023 talk or 2022 he is quoting 1927 Fritz London!! wow.
London F 1945 Rev. Mod. Phys. 17 310
So in 1927 Fritz London says it's not the magnitude of the momentum but the direction (as time) that is changing... that is precisely the noncommutative secret!! Or conversely the "spacetime" as relativistic gauge depends on the magnitude but NOT the direction.
If one wishes to assume that it has any influence on the gauge at all then it can depend upon only the magnitude of the four-dimensional displacement of the line segment, but not on its direction.Basil J. Hiley in 2016:
However we are talking about a single particle, and therefore such an approach..........
requires the existence of some form of sub-quantum medium, an old idea which has dropped out of favour. In our approach the two momenta [left and right direction] arise because the order of succession is vital and gives rise to a non-commutative structure. In such a structure it is necessary to distinguish between left and right translations and it is this difference that gives rise to the ‘strange’ features of quantum processes.
New Concept of Motion
Notice that in a non-commutative structure, a key factor is the recognition that we must
distinguish between left and right translations. In other words, order is vital. The distinction
gives rise to the possibility of a new type of motion, namely, that an old structure may evolve into a new structure via the inner automorphism
OK so Hiley goes into more details on Heisenberg here! fascinating.
Now we are in a position to explain the standard ket, a new notation first introduced by
Dirac [8] and subsequently discussed in more detail in the 3rd edition of his classic text on
quantum mechanics [12].
Only multiplication from the left is allowed. He called the object ⟩ the standard ket and its dual ⟨ the standard bra. We then have a way of distinguishing between left and right translations, a distinction that is essential in non-commutative geometry.
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