This website, www.Snuark.com is about snuarks. This website is only a few days old. Soon enough it will be connected to LaTeX formatted acrobat files that will explain this far better than this html. But for now, here are some simple definitions.

What is a snuark? Well it's a subparticle of a quark or lepton. Are snuarks the ultimate smallest elementy of matter? Not quite, there is another level smaller, the binon. Mathematically, snuarks and binons are elements of a Clifford Algebra. Clifford algebras are generalizations of the Pauli and Dirac algebras that are used to model spin-1/2 and relativistic spin-1/2, respectively.

Binons are primitive idempotents. "Idempotent" means that they remain unchanged when squared. In the real numbers, the only two elements that are idempotent are 0 and 1. In more complicated algebras, the idempotents have a more complicated structure. "Primitve" means that they cannot be written as a sum of two idempotents. Among the reals, 1 is primitive, but 0 is not primitive because 0+0 is 0.

Among the Pauli algebra, the binons are simply the spin projection operators. The Pauli spin projection operators all have a trace of 1. They can be written as (1+sigma)/2 where "sigma" is the spin operator in some direction. Among the Dirac algebra, the binons are the projection operators that have a trave of 1.

It is well known that velocity operator of the Dirac equation has eigenvalues of +-1 (or +- c). When the Dirac algebra is generalized to a larger Clifford algebra, a new velocity operator is produced. The more general velocity Dirac velocity operator includes the possibility of having velocity in a hidden dimension but we will restrict our attention to velocity operators that correspond to velocities in the usual 3 dimensions only. Even in the generalized Clifford algebras, these velocity operators are the same as those of the usual Dirac equation.

A snuark is a collection of orthogonal binons that all travel in the same direction. "Orthogonal" means that they multiply to zero. For example, the projection operators for spin-1/2 in the +z and -z directions are orthogonal. That the binons travel in the same direction is equivalent to saying that they are in the same velocity ideal.

For example, the Dirac velocity operator for velocity in the +z direction is defined by
.
and the binons travelling in the +z direction must live in the ideal generated by the projection operator for velocity in the +z direction:
.

What binons could make up such a snuark? To answer this question first we must find the binons that lie within the ideal generated by this velocity. After studying a little Clifford algebra, we find that the general form for the binons compatible with this direction is of the form:

where a^2+b^2+c^2 = 1.

Any binon of the above forms, by itself, could be a snuark for the +z direction in the Dirac algebra. To get a snuark made up of more than one binon, we have to choose two that are orthogonal. If the first binon is defined by (a,b,c), the second must be defined by (-a,-b,-c).

That's all there is to snuarks in the Dirac algebra. The Dirac algebra is big enough to hold electrons and positrons, or neutrinos and antineutrinos, but it isn't big enough to hold both the charged and neutral leptons. To get more general relations between the particles we have to go to a bigger Clifford algebra. It is there where snuarks become less trivial.

I think this is enough for now. I'm busily working on the paper that covers this subject more deeply. Come back in a few months. My paper will be out, and I'll type up some better educational material.