Crazy Coincidence about alpha

[Martin]

http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant.html

 

 

alpha, approximately 1/137, is maybe the most important pure number in quantum theory of any kind, at least in QED

and Severian disagrees

 

and indeed alpha changes at very short distances and high energies but so what? in the ordinary world of atoms, molecules, and even nuclei alpha is ordinary alpha---and it is basic to everything around us.

 

so feynmann, among others, said every physicist worth his salt should have that number tacked up on the wall to remind him to ask why it was what it was.

 

alpha is only known by experimental measurement---basically by measuring the strength of electromagnetic interactions, the effects of the strength of the EM coupling. It is an unexplained number in nature.

 

When people produce mathematical "formulas" for alpha it is usually considered to be a mild form of insanity. alpha is a number that permeates the world appearing everywhere and in everything but you must not pretend you have a formula for it, or people will think you are strange.

 

Hans DeVries has a formula for alpha.

 

think of it as a wild coincidence that it works. or maybe it doesnt work, there is always that hope. I will copy Hans formula here to take a closer look at it.

[Martin]

First of all, I recall that Feynmann was just as interested in the square root. It is just as basic. So we can be interested not in the number 137.0359991 ( which is the experimentally determined value now) but in its square root 11.70623762...

 

Let us define a function that will give us this 11.706....thingee

 

Define F(X) this way:

 

[math] F(X) = X +{X^{-1} \over (2\pi)^0 }+{X^{-3} \over (2\pi)^1 }+{X^{-5} \over (2\pi)^3 } + {X^{-7} \over (2\pi)^6 } +....[/math]

 

Then, Hans de Vries says, if we solve the equation

 

[math] F(X) = e^{\pi^{2}/4}[/math]

 

the solution X will turn out to

be this famous universal number 11.70623762...

which so far has only been determined by experimental measurement.

 

is this sheer coincidence, or does it seem to you that there might be a physical reason for it?

 

the series 0,1,3,6,10,15

is the series of triangular numbers you see in a "bowling pin" setup.

0, 0+1, 0+1+2, 0+1+2+3, 0+1+2+3+4, ... and so on.

[Severian]

Interesting, but I fail to see why 1/137.035... would be fundamental. It is only the asymptotic limit of the QED beta function. As I have pointed out before, alpha changes with energy so it is not 1/137 for most energies. 1/137 is just the value at really low energies.

 

I imagine that if you work hard enough, you can find an equation which gives you any number you like. It would be interesting to know how far in decimal places the correspondence goes...

 

PS: Are you sure you got F(X) correct? It is a little odd that the first two entries both have no pi in them. I would have expected the first term to be [math]2 \pi X[/math].

 

PPS: It is also interesting how F(X) seems to change as the inverse square of X (each term is X^-2 times the previous one). According to the thesis, X^-2 would be alpha itself....

[Martin]

... It would be interesting to know how far in decimal places the correspondence goes...

 

...

 

Some checking was done this morning' date=' Severian.

The correspondence goes out 10 decimal places.

 

that is, the formula gives agreement with the experimentally measured

137.03599911 (from the NIST website, 2002 CODATA rec. value)

to ten significant figures.

 

It is quite a bizarre coincidence don't you think?

 

Here are the essential steps in a calculation used to check it.

 

Define F(X) this way

 

[math'] F(X) = X(1 +{X^{-2} \over (2\pi)^0 }+{X^{-4} \over (2\pi)^1 }+{X^{-6} \over (2\pi)^3 } )[/math]

 

this way all the powers of X are even and one can use 137.03599911, reducing the amount of arithmetic, then at the end there is one multiplication by 11.70623762

 

The idea is plugging in the experimenally measured value for X one should get (to a good approximation) the exp(pi^2/4) number, which is 11.79176139...

 

the coincidence(?) found by Hans de Vries is:

 

[math] F(X) = e^{\pi^{2}/4}[/math]

 

So, according to this, one expects

 

[math] F(11.70623762...) = 11.79176139...[/math]

 

Plugging 11.70623762 into the function F(X) defined here, and calculating, one gets 11.79176139.

 

But this is also equal, to the exp(pi^2/4) number, to ten decimal places, namely to 11.79176139.

 

The original F(X) function was this power series

 

[math] F(X) = X +{X^{-1} \over (2\pi)^0 }+{X^{-3} \over (2\pi)^1 }+{X^{-5} \over (2\pi)^3 } + {X^{-7} \over (2\pi)^6 } +....[/math]

 

but I truncated it when the terms got ridiculously small, and factored out the X.