de Vries: Spinors after all?

From: Alejandro (arivero_at_posta.unizar.es)
Date: 10/25/04 

Date: Mon, 25 Oct 2004 13:09:40 +0000 (UTC)

Funny enough, it seems that HdV formulas could hide spinors after all.
In a third set of guesses, Hans de Vries has pointed out another
mnemotechnic for leptons. But this one asks for the Pythagorean
relationship, which as we know is the hallmark of spinors. They should
be Pythagorean Spinors ( http://math.ucr.edu/home/baez/week196.html ),
because they are integers.

Point is, consider the only two triples having consecutive integers,
this is, (3,4,5) and (-1,0,1). Given any three numbers a+b=c, we can
use them to build a unit-independent relation between quantities x, y
,z:
                  a ln x + b ln y = c ln z
ie, if we change units x --> k x, y -->k y, z--> k z, the relation
still holds.

Now we used both triples, in a not very straightforward way, to build
one relation of this kind:

   (3^2+0^2) ln x + (4^2 + (-1)^2) ln y = (5^2 + 1^2) ln z

 Which holds very fairly when [x,y,z] = mass of [electron, tau, muon]
respectively.

With a bit of algebraic manipulation the equation can be written

ln z/y
--------- = 3^2 / (4^2+1)
ln x/z

and then we can compare it with the same quotient from the first set
of HdV guesses:

ln z/y pi^2 -1
-------- = --------------
ln x/z pi^2 -3

On a first view, it seems a failure. But noticing that pi^2 can be
expanded as one sixth of the sum of inverse squares (Zeta(2), so to
say), we see that the first term of this expansion is not far away
from the one in the new guess. In fact they coincide if we "correct" a
bit the formula to be expanded:

ln z/y pi^2 -1 -1/2
-------- \approx ---------------------
ln x/z pi^2 -3 -1/2

which makes oneself to feel more motivated about the existence of a
meaning for this family of approximations.