Fine Structure Constant

Question

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.

Answer 1

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.