Natural Science Forum / Physics / Particle Physics / June 2007

There is an amusing way to relate the Standard Model families to a
(1,3) geometric system. Consider dt, dx, dy, dz and then put charges:

dt is to be colour neutral, electric +1
dx, dy, dz are to be coloured (r,g,b respectively, say) and electric
-1/3

then the exterior algebra components, which are

1
dt, dx, dy, dz
dt^dx,   dt^dy,   dt^dz,   dx^dy,   dx^dz,   dy^dz
dt^dx^dy,   dt^dx^dz,   dt^dy^dz,   dx^dy^dz
dt^dx^dy^dz

come to have the following combinations of colour and electric charge:
__
_+1, r-1/3, g-1/3, b-1/3
r+2/3, g+2/3, b+2/3, rg-2/3, rb-2/3, gb-2/3
rg+1/3, rb+1/3, gb+1/3,  _-1
__

which are, I do note need to tell you, the charges of the elementary
fermions in the standard model.

Furthermore, the intersection product with the volume form can be used
to map a combination to its oposite,
eg:  dx * (dt^dx^dy^dz) =  dt^dy^dz
(ie dx has charge  [r,-1/3], and maps to dt^dy^dz which has charge
[ gb, +1/3  ]: its antiparticle )

I am pretty sure this structure has been noticed before both in the
alternative literature (including, but not only, quaternionic games)
and in the standard one, and I would like to collect references to it.

Note also that if we add a neutral fith-dimension, then we can group
the particles in the way they are grouped in SU(5) representations: a
1 plus a 5 plus a 10. But then we need to consider grading. This is
interesting by itself: it could be used to consider generations, or to
create supersymmetry.

Aside: I remember when I was younger I tried to relate the SM to a
metric by postulating that the confined coordinates -quarks- were
angular variables, while the leptons were the variables for radius and
time, in a Schwarchildian metric. I am surprised how I missed this
more elementary idea of having the positron in the time variable and
thre three colours of the down quark in the role of spatial variables.
It is elegant, and easier to swallow once one notices how different
are the mass mechanisms of the neutrino and the up family.