Is there any reason that in principle the basis of the 'fine structure constant' ~[1/137.036] cannot be understood as a simple geometric ratio?

For example, commencing with a [2 x 2 x 2] unit cube, centre [N], of 1/2-pole [root3]=[*3] and 1/2-face diagonal [root2]=[*2], and imagining 3 fundamental components [A]=[2--*3], [B]=[2--*2] and [C]=[B--A]=[*3--*2]. then if the component [A] is projected from the pole to the face diagonal, it becomes ([*3] x [A])/[*2]=0.3281694=[Ap]; such that, relative to [C]=0.317837245196, a disparity [p]=0.0103321540278 is implied: ([Ap]--[C])=[p].

Now if these components [Ap] and [C] are permitted to elaborate on this common face diagonal axis of the cube such that multiples of [p], thus [2p], [3p] and so on arise to define a cumulative disparity between them, one immediately notices that 137p=[*2+p'], where [p']=0.00129153943, and [8p']~=[p], or equally, [*2]/[p]=136.875.

If it is further supposed that [*2] represents an effective limit in this divergence between [Ap] and [C], following upon which in a sequential linear elaboration of these components in an extrapolation of that axis in [*2] unit intervals, these then converge, and if with respect to that first cube and a cubic lattice structure extrapolating from it a second cube and lattice identical to it is imagined whose corner is [N], then if an oscillatory dynamic is imagined to inhere between these two aspects of what becomes a 'reciprocal cubic lattice structure' in space --potentially a 'unitary phase structure'--, these components [A], [Ap] and [C] arising in its poles and (horizontal) diagonal face axes in particular may be considered to define the basic configuration of such an interplay; whereupon the residual [p] and the ratio [p]/[*2] become definitive and central.

Moreover, if this dynamic interplay is considered within a singular context--even a universal frame in which it is mediated by a correspondingly singular 'force'--, the basis is also conceivably suggested for a 'unitary wave principle' primarily comprising these two principal components. As such, it is also worth observing that the relative frequency of occurrence of [p] and [p'] is precisely [137:1]. The question stands.


closed as off-topic by John Rennie, Kyle Kanos, stafusa, Daniel Griscom, Jon Custer Oct 30 '17 at 13:04

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    $\begingroup$ The fine structure constant is definitely not exactly 1/137 or 1/136.875. it is in fact 1/137.035999173 to an accuracy of about 0.25 parts per billion. If you wish your exercise in geometry to be taken seriously as the source of the fine structure constant then you must furnish here a compelling and fundamental explanation of exactly why this must be, an accounting of why your result differs from this one, and finally why the fact that it matches the first three significant digits of the real value for the constant is not merely a coincidence. $\endgroup$ – niels nielsen Oct 29 '17 at 6:29
  • $\begingroup$ Related: physics.stackexchange.com/q/134802/2451 and links therein. $\endgroup$ – Qmechanic Oct 29 '17 at 16:09
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    $\begingroup$ This site uses LaTeX notation for mathematics; please use that to format your mathematics. As it stands your post is extremely hard to read. $\endgroup$ – Emilio Pisanty Oct 29 '17 at 16:20
  • $\begingroup$ Obviously a comprehensive explanation is necessary for the difference between the substrate ratio suggested, [p]/[*2] or [p]/[sqrt 2], and the value measured at the QED energy scale. This explanation however is far too complex to include here and depends on the 'mathematical harmonics' of the geometric model of a unitary space in which the relation of the basic wave component or 'axial vector' [C] to the same 'harmonic interval' [*2] is definitive, if you will permit such language. $\endgroup$ – jeremiah Oct 30 '17 at 6:19

The couplings in quantum field theory depend on the energy scale (see: "running coupling constant"). The value that is normally quoted for the fine-structure constant, approximately 1/137, is only the value at low energies.

If the various couplings are extrapolated to high energies, around 10^10 GeV, they almost converge on a common value. This is a major motivation for "grand unified theories", according to which there is one unified force (this does not include gravity) which is then broken into several different forces by a heavier version of the Higgs mechanism.

In string theory, such forces are often described by open strings attached to a brane, and the strength of the coupling depends on the volume of the brane. In an advanced version of "F-theory" (a form of string theory), one may find branes which are made of a finite number of "fuzzy points"; and in one type of model, a brane made of 24 or 25 of these fuzzy points, implies a coupling for the grand unified force of 1/24 or 1/25. This is actually the right magnitude for grand unification of forces at high energy scales; so the fine-structure constant of QED, along with the couplings of the other forces, would then result from applying the Higgs mechanism, and then running this number 1/24 down to low energies.

All that is the only example known to me, of a functioning quantum field theory in which the fine-structure constant really is explained as a simple ratio. By a "functioning quantum field theory", I mean a theory which can produce calculations like those employed in quantum mechanics and particle physics. Many many people have proposed so-called "numerological" formulas for the fine-structure constant, but none of those formulas are part of a functioning conceptual and calculative framework, that can explain the behavior of actual photons and electrons.

It is possible that some of these numerological formulas could nonetheless be given life in a proper field-theoretic framework; but here the main problem I already mentioned - people try to explain this number 1/137, but in our current understanding, the more fundamental value (that the theory should explain "directly") is the high-energy value. Quantum field theory does contain the concept of an "infrared fixed point", in which the low-energy running converges on a specific value; so maybe one could justify one of the formulas for 1/137 in that context. In that regard, my favorite observation (due to Vladimir Manasson and Mario Hieb) is that the fine-structure constant equals 1/(2pi times Feigenbaum's constant squared). Manasson has written a few papers trying to make a theory out of that observation.

So to sum up, I don't say it's impossible that there is a well-defined theory in which the fine-structure constant has its value because of a geometric relationship like the one that you name; but it's difficult because coupling constants run with energy, and one does not expect this most-quoted value of the electromagnetic coupling to be the truly fundamental one.

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    $\begingroup$ Suggestion: before jumping straight to string theory, perhaps you could point out the measured value of $\alpha$ at the Z pole, 127.916 ± 0.015, to illustrate how it changes (source: PDG) $\endgroup$ – user154997 Oct 29 '17 at 10:08
  • $\begingroup$ The geometric model of a 'unitary universal field' in which this substrate ratio [p]/[*2] is central--and in which the correction required to that ratio to imply the 'fine structure constant' is implicit-- is primarily a model argued as capable of explaining phenomena such as the behaviour of theoretical electrons and photons at low energies: it purports to explain the 'physical basis of QED theory' itself. On the presumption that the SM developed at higher energies is only an extrapolation of QED, such a model is also potentially capable of explaining observations at those energies. $\endgroup$ – jeremiah Oct 30 '17 at 6:36

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