I'm getting a paper ready for submission to Foundations of Physics on
the subject of
spin-1/2 and the generations of elementary fermions. The paper uses
geometric algebra
but it's written in the language of the Pauli algebra; I decided the
audience would be
larger for a paper written that way. So instead of using spinors, it
describes spin
states with projection operators such as (1+sigma_z)/2 for spin-up:
http://www.brannenworks.com/Gravity/EmergSpin.pdf
The basic idea is "can we make spin-1/2 act like position?" I.e. if
you measure the
position of an electron you will disturb its momentum and if you
measure its position
again you will get a very much different result. But if you measure
spin twice you will
get the same result.
Classically, when you measure spin you get the axis and amount of
spin. With spin-1/2,
you choose the axis, say z-axis and get only +h-bar/2 or -h-bar/2
which correspond to
spin in the +- z directions.. The proposed spin is midway between
these. Instead of
choosing an axis you choose a coordinate frame and in addition to
getting +h-bar/2 or
-h-bar/2 you also get which of the three axes the spin is measured
with respect to.
This has to do with the theory of "mutually unbiased bases" or
"incompatible operators"
taken over a finite Hilbert space.
Anyway, the paper resums the path integrals over the modified spin
space and
shows that you regain the usual spin-1/2. In addition, you get the
generations
of elementary fermions. Doing the resummation the usual way would have
been
horrendous, but with geometric algebra it is not so bad.
I'm figuring on submitting the paper in about a week, any suggestions
before then
would be much appreciated.