# Solution to a 20th Century Mystery

## 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(n1, n2)} dependent on the integers n1 n2 as

α(n1, n2) = α(n1, ∞) π(n1× n2)/π, ...........................†

where

α(n1,∞) = cos(π/n1)/n1.

The numerical value of α, the fine structure constant, is given by the special case n1 = 137, n2 = 29.

Thus

α = α(137,29) = 0.0072973525318...

The experimental value for α is

αexp = 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.