Infinite Dimensionality as a Bridge to Quantum Mechanics
In Physics, more specifically in Quantum Chromodynamics there are neat renormalization tricks, meant to ABRACADABRA some infinities (self-energy) etc...
One of these tricks is called Dimensional Renormalization. Normally it involves redeploying a theory in an N dimensional space and then making N go to 4 or 3. When that happens the infinities disappears or become meaningful within that new framework.
Recently, Dudley R. Herschbach, A.K.A. dude...:) (just kidding, Dudley) used a variation of Dimensional Renormalization in which instead of recovering regularized equations when N goes to 4, he obtained "Surprisingly Accurate Semiclassical Results" resembling Pure Quantum Mechanical Results by making N go to infinity...:)
Also surprisingly is that that same trick worked fine in the calculation of energy levels of atoms (molecules with a single positive center..:)
That is fine and dandy...:) but what is missing is the reasoning behind that coincidence...:) Nobody, including Dudley, is saying that the Universe is infinitely dimensional or the reason for Quantum Mechanics is the infinite dimensionality of the Universe. I should comment that he used the word Dimensional Scaling as opposed to Dimensional Renormalization, but that is a minor detail, just a name...
The unaddressed Hocus Pocus is the WHY? Why would an infinite dimensional Universe lead to Simple and Surprisingly Accurate Approach to the Chemical Bond Obtained from Dimensional Scaling ...:)
To understand that, one should delve deep into the details of the derivation by Professor Herschbach. He did not found that any kind of infinitely dimensional topology yields Accurate Results in the case of Diatomic Molecules (e.g. H2) ...:)
Only a special topology in which rho is infinitely dimensional, Phy is the dihedral angle between the planes containing the electrons and the internuclear axis of a H2 molecule. This special topology makes Rho perpendicular to the interatomic axis of H2.
If you remember that Quantum Lagrangian Principle (QLP), the basis for my Grand Unification Equations derivation, you would realize that that choice of topology is a requirement and the WHY for the Surprising Success of Infinite Dimensionality Scaling.
If you read equation (6) and (7) and understand Figure 2 on his paper, it should become clear that the difference between the incorrect equation (7) and the correct one (6) is the loss of dependence on Phy. This can be seen as if the two Hydrogen nuclei were generating cylindrical dilaton waves and that the electron wavefunction had a kr component with a de Broglie wavelength r1 and r2. This is exactly what one would expect from the Quantum Lagrangian Principle.
IF electron 1 is at distance r1 from nucleus 1, then it is an integer number of wavelengths from that nucleus or conversely, the de Broglie wavelength (5D dilaton projection onto the Hyperbolic 4D spacetime) is the distance to the nucleus divided by an integer n1. This reasoning leads to the proper calculation of excited states.
Remember that the de Broglie wavelegth associated with the nuclei is much broader due to their 2000 times larger inertia. As you remember, the fundamental dilator has different inertia for the different phases of the coherence.
The QLP states that a dilator will always dilate in phase with the surroundings dilators. This works both in the 5D spacetime as well as in the 4D spacetime projection. The 5D dilatons project onto a 4D Hyperbolic Hypersurface as de Broglie Waves. This means that the Hydrogen atoms will be sitting on the crest of the electron de Broglie wave (at a given Rho-directed momentum).
I wrote this some time ago. I don't have time to review it, but it is likely to be correct. Please review the reasoning and comment on it. You might have to read Dudley's paper.
Cheers,
MP
PS- I found this laying around and thought it would be worthwhile publishing. I was planning to show the detailed derivation of Quantum Mechanics (Schrodinger Equation) from the Quantum Lagrangian Principle, which means a renormalization of the dynamics from 5D Spacetime to 4D Spacetime. The surfing the dilaton field aspect of the Quantum Lagrangian Principle brings in the ondulatory aspects of Quantum Mechanics. The reduction of one dimension introduces the uncertainty in quantum mechanics - the interference between dilaton fields produces lines in our 3D space where the dilator can be at each step of the hyperspherical expansion. This results uncertainty on the path of a dilator (electrons). The loci of these positions is described by a dilator wavefunction...:)
I will get back to this problem later. Let me know if interested in helping me. I've been very busy and it was never my idea to write the whole chimichanga. By the way, what I wrote above should be self-evident...:)
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