# Solution to a 20th Century Mystery

## Feynman's conjecture of a relation between α,
the fine structure constant, and π

## James G. Gilson, j.g.gilson@qmul.ac.uk

### Feynman's Conjecture

A general connection of the quantum coupling constants with π was anticipated by R. P. Feynman in a remarkable intuitional leap some 40 years ago as can be seen from the following much quoted extract from one of Feynman's books.

*There is a most profound and beautiful question associated with the observed coupling constant e the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
*
### The Solution

It will here be shown that this problem has a remarkably simple solution confirming Feynman's conjecture. Let P(n) be the perimeter length of an n sided polygon and r(n) be the distance from its centre to the centre of a side. In analogy with the definition of π = C/2r we can define an integer dependent generalization, π(n), of π as

π(n) = P(n)/(2r(n)) = n tan(π/n).

Let us define a set of constants {α(n_{1}, n_{2})} dependent on the integers n_{1} n_{2} as

α(n_{1}, n_{2}) = α(n_{1}, ∞) π(n_{1}× n_{2})/π, ...........................†

where

α(n_{1},∞) = cos(π/n_{1})/n_{1}.

The numerical value of α, the fine structure constant, is given by the special case n_{1} = 137, n_{2} = 29.

Thus

α = α(137,29) = 0.0072973525318...

The experimental value for α is

α_{exp} = 0.007297352533(27),

the (27) is ± the experimental uncertainty in the last two digits.

The very simple relation † between α, the fine structure constant, π and π(n) confirms Feynman's conjecture and also his amazing intuitional skills.

### Link

For details of how the formula † was obtained and some of the consequences arising from it visit the website:-

http://www.fine-structure-constant.org/

Reply to Oneiromancer: There have been dissenters from every theoeretical physics construction ever published. However,
in a public forum, cogent arguments are more to be desired than a case resting on *I don't consider............ *

You suggest that if the electromagnetic force is geometrical then, the other forces should have the same value. This is equivalent to saying that all polygons should have the same size and the same number of sides. In fact, if you look at equation † again, you will see that it expresses a proportionality between values of α like representations and π like representations. This means that geometry alone may not be all that is involved. However, it seems to me, not to matter if *only* geometry is involved. After all, what is general relativity if not just geometry? The integers 137 and 29 are quantum numbers specific values of the pair (n_{1}, n_{2}). Other specific values of this pair of quantum numbers correspond to other values of the fine structure constant at different transfer energies or to other coupling constants. α(29, 137) for example, gives the value of the electroweak coupling constant. Visit my website to find out more about the details of the theory.