To see some possible wave capture diagramatic representations for values of the strong coupling constant click the fine structure constant logo on the right.A general formula for possible numerical values for quantum coupling constants comes out of this vacuum polarization theory. The formula depends on two integer parameters n1 and n2 and can essentially be read off the wave capture diagram which as given on the previous page represents the special cases n1 = N unspecified, but fixed n2 = 3.
The formula is
α(n1,n2) = n2cos(π/n1)tan(π/(n1× n2))/π.
The values of the quantum coupling constants are often denoted by the lower case Greek letter alpha, α with some specialising subscript. The fine structure constant itself is usually denoted by alpha with no subscript.
The numerical value of the fine structure constant α is given by the special
case n1 = 137 with n2 = 29,
α = α(137,29) = 29cos(π/137)tan(π/(137×29))/π = 0.00729735253186... .
The latest CODATA recommended experimental value for this quantity with the (± 27) uncertainty range centered on last two digits (33) is,
α = 0.007297352533(27).
I formally predicted the theoretical value given above for α = α(137,29) with an announcement in The Times Newspaper of July 21st 1999, preceding CODATA's publication of their recommended experimental value.
There is a very specific place taken by π in the formula for α. and indeed π plays a central part in all of this theory. A connection of α with π was suggested by R. P. Feynman in a remarkable intuitive leap some 40 years ago.
The numerical value of the electro-weak coupling constant αg is given by the special case n1 = 29 with n2 = 137,
αg = α(29,137) = 137cos(π/29)tan(π/(137×29))/π = 0.0342806263570... .
The numerical value of the strong coupling constant αs at the energy of the tau-meson is given by,
αs(mτ) = α(2,1) = 1/π = 0.318309886183... .
To see some possible wave capture diagramatic representations for values of the strong coupling constant click the fine structure constant logo on the right.
Check out the latest recommended values for the fundamental physical
constants at CODATA web site
and see gilg.pdf,
gilj.pdf, gil2.pdf and gil3.pdf.
Inspection of the wave capture diagram reveals that the structure is part of an equilateral n1n2 sided polygon, the part displayed being the location of the n2 part angular segmented trapped wave between a and d. Such a polygon can also be used as a representation of a cyclic group of order n1n2. This holds the implication that the values of the coupling constants are related to the n1n2 order cyclical group. It turns out that separate subgroups forming the product group of orders n1 and n2, have characterisation properties that coincide with the numerical values of the two types of coupling constant such as α and αg. This pairing up of the two types of coupling constant makes possible the identification of running coupling constant pairs right up to the grand unification energy. See gil2.pdf.
A number of results in high energy physics that come out of this structure include a very simple theoretical formula giving a generalization θG(n1,n2) of Weinberg's weak mixing angle θW
sin2(θG(n1,n2))= n2cos(π/n1)/(n1cos(π/n2)).
An energy running version of this quantity can also be formed. The low energy value is given by,
sin2(θG(137,29)) = 0.21287103846... .
The ratio of the rest mass of the W particle to the rest mass of the Z particle comes out as,
mW/mZ = cos(θG(137,29)) = 0.887... .
The experimental value for this quantity is 0.881. See gil2.pdf and gil3.pdf.
Finally, the formula for the fine structure constant can be used to set up a scheme for the finite renormalization of Quantum Electrodynamics. This makes it possible to avoid the mathematical uncertainties associated with the manipulations of infinite quantities that have hitherto been necessary in the renormalization proceedure. See gil1.pdf.
----------------------------------------Back To Page 1.
You may have to download the utf-8 character set from Alan Wood or from Unicode Consortium in order to read these pages and download a free copy of Adobe Acrobat Reader in order to read the gil*.pdf files.