The Model

James N. Bellinger

Atoms are made of a nucleus and electrons; nuclei are made of protons and neutrons, and these are made of quarks and gluons. We can ask if the quarks (and electrons) are themselves made of something even smaller, which we usually call preons [pree-on].

My model has 7 assumptions.

picture of merging particles picture of a particle decaying
  1. All particles are composed of combinations of preons.
  2. All interactions of preons are 3-body interactions. Notice that if we find particle A and particle B combining to form C , we are required to suppose that we can have particle C decomposing into A and B.
  3. I ignore position and momentum. The particles are close in some sense, of course.
  4. Every preon may interact with every other preon to make a new preon. This means that the preons cannot have any net charge!
  5. We have some preon I which has the property that it can be absorbed or emitted from any other preon without changing the nature of that preon. This is like an electron absorbing a photon: the electron is still an electron. picture of a particle emitting another
  6. If we have three preons interacting one after another, it doesn't matter whether we group the first two interactions together or the second two--we get the same result. (A+B)+C == A+(B+C)
  7. Every preon has an anti-preon. Combining the preon and its anti-preon results in the special preon I which I mentioned above.

The model described above corresponds to what mathematicians call a Group; and since I only expect a handful of preons, this is a Finite Group. These are generally well understood, and the continuous symmetries I find can be explained in terms of their representation theory. Unfortunately the theory requires a good deal of background, and the representations handiest for displaying the symmetry obscure the possible physics content.

To study the symmetries I decided to treat sums of group elements as vectors and look at the ways matrix transformations could mix them up without changing the essential nature of the group. The matrix transformation turns a group element into a sum of various group elements. I required that the collection of sums of group elements you get after the transformation of the group elements obeys the same group multiplication rules as did the original set. Put simply, the transformation can mix things up but the group better still work. This turns out to be a pretty stiff requirement.

James N. Bellinger U. Wisconsin at Madison Physics Dept.