"Bohm (and also de Broglie (1960)) called this term the “quantum potential energy.” This new quality of energy enters as the coefficient of h2 [relativistic from de Broglie] and this is why Dirac missed the QHJ equation. Its appearance is intimately connected with the Baker bracket (Jordan product) and therefore the non-commutativity of (x✭ p)."
https://royalsoc.org.au/images/pdf/journal/154-2-Hiley.pdf
Thus we see that, mathematically, the quantum potential arises as a consequence of the difference between the mean of the square of the momentum and the mean momentum squared. All this implies that the dispersion in the momentum for a single particle in quantum mechanics will, in general, be nonzero. For the single particle in classical physics the momentum is always dispersion free. In this way we see that the ✭-product contains the structure that guarantees the existence of the uncertainty principle, contrary to what Dirac claims.
https://www.youtube.com/watch?v=jl00BY8kopw
"Further, by emphasising the constancy of h-bar in the relation [change of]X times [change of]P approximates h-bar one tends to be led to the notion that the 'disturbance' is dependent only on the size of the cell in phase space. In this way the overall experimental conditions were tacitly dismissed as irrelevant."
"There is a deeper [noncommutative] structure that gives rise to a noncommutative phase space...momentum and position."
the Jordan product becomes the normal inner [matrix cross] product. In
symbols (AB + BA)/2 —> AB. The Jordan product is the most neglected productin the whole discussion of the foundations of quantum mechanics.
So there's the Baker bracket [Jordan product as the noncommutative average] that gives the Bohm Momentum [trajectory]....
https://www.youtube.com/watch?v=q_jHmoxuxsY
I'm going to give him extra credit now, what de Broglie said was the appearance of the quantum potential makes it look as if the mass has changed i've had to put squares here [Planck's Constant] because i'm going to relativity just don't worry about it there's a reason for doing this the point is the rest mass has changed and that's exactly what Feynman did and de Broglie was told to, in no uncertain terms, to drop dead which he did very shortly afterwards
Journal and Proceedings of the Royal Society of New South WalesJournal Article December 2021 The Moyal-Dirac controversy revisited Authors: B J Hiley
To repeat, it is only when we go to order O(h2) [Planck's constant squared] and above that the Baker bracket [Jordan Product] does not reduce to the usual commutative product. Generally terms of O(h2) are assumed to be negligible and therefore are not discussed, but the bracket plays an important role when energy (Hiley 2015) is involved. A careful study of Pauli’s (1926) application of the algebraic approach to the energy level structure of the hydrogen atom shows how a Jordan product enters into the calculation. As we have already pointed out, one of the advantages of the Moyal approach is that it contains classical physics as a limiting case as is clearly seen from equation (12)....
There is no need to look for a one-to-one correspondence between commutator
brackets and Poisson brackets, a process which fails as was demonstrated by the well-known Groenewold-van Hove “no-go” theorem (Guillemin and Sternberg 1984).
https://arxiv.org/pdf/1206.3116.pdf
On the other hand, it is known that there is no such globally defined vector field on the 2-sphere. (In fact, among the closed 2-dimensional surfaces only the torus has one.) The way around this difficulty is to complexify the (tangent bundle of the) phase space.
Groenewold proved that Weyl's and von Neumann's misplaced expectations of such were off. So, more so than in the past, people now quantize heuristically, pulling quantum theories out of well-informed hats, and are at peace with it.)
Groenewold's correspondence principle theorem enunciates that, in general, there is no invertible linear map from all functions of phase space
to hermitean operators in Hilbert space , such that the PB structure is preserved,as envisioned in Dirac's functor heuristics.
https://arxiv.org/pdf/1506.08203.pdf We analyze the no-cloning theorem in quantum mechanics through the lens of the proposed ER=EPR (Einstein-Rosen = Einstein-Podolsky-Rosen) duality between entanglement and worm-holes. In particular, we find that the no-cloning theorem is dual on the gravity side to the no-go theorem for topology change, violating the axioms of which allows for wormhole stabilization and causality violation. Such a duality between important no-go theorems elucidates the proposed connection between spacetime geometry and quantum entanglement....As we have seen, spacetime topology change leads inexorably to violation of causality, via either breakdown of the Hausdorff condition or creation of traversable wormholes. Using ER=EPR to translate this result to quantum mechanics, we find that violation of the axioms of the topology-conservation theorems is dual to violation of monogamy of entanglement (i.e., cloning) and the existence of wormholes is dual to the existence of entanglement entropy....It is striking that on both the general relativistic and quantum mechanical sides of the duality, violation of the no-go theorem leads to problems for causality. The unexpected connection between cloning and topology change offers support for the ER=EPR correspondence, which provides a natural explanation for their relation.
