Kea said:
CarlB, please give our readers a summary of your argument.
The idea behind "particle physics" is that as a physical object is made more and more energetic, it becomes simpler because it is broken into smaller and smaller parts that can be treated alone. For example, on a cold morning your car might have extravagant patterns of frost on it. As your car warms up, these change to dewdrops, which are simpler because the water molecules aren't stuck together any more. Warming further, the dewdrops evaporate into the air, which is simpler yet. Heat that water up higher, it disassociates into atoms. Hotter, and it becomes nuclei and electrons. Hotter yet, and you get neutrons, protons and electrons. Hotter still, and you presumably get up and down quarks and electrons. As the material gets hotter, it splits into simpler subparts, conversely, as it cools, it condenses back again.
But at the level of the electrons and up and down quarks, the splitting/condensation sequence somehow gets complicated. As you add more energy to the system, instead of continuing to break into smaller parts, one finds that these particles are "point particles", and in adding energy one must suddenly deal with a whole plethora of unusual particles from the other two generations. The point here is that this is contrary to our expectations that things should get simpler as we heat them up. Instead, what is happening is that the extra energy allows new degrees of freedom to show up.
I assume that the three generations of elementary fermions must be made up of simpler particles, or "preons". Lee and Sundance's idea is similar in that they also assume preons, and their preons have some similarities to mine in that both ideas assume that the quarks and leptons are unified and are composites more or less formed from triples of subparticles.
Their idea is that the quarks and leptons are each made up of three subparticles which they call Helons, H_0, H_+, H_-, where the suffix gives the electric charge in units of 1/3 e. These are braid groups, more or less. The three generations arise from differing complexity in the number of crossings of the braid group. They are unable to explain why there are exactly three generations or why they have the relative masses.
I model the subparticles based on a type of Clifford algebra called a "geometric algebra", a theory which was started by David Hestenes 25 years ago. This is different from braids in that it is a heck of a lot simpler to do calculations. If you don't want to learn a bunch of math, you can think of a great example of a Geometric Algebra as being the set of functions that map from space time to the set of 4x4 matrices that you could describe by taking sums of products of Dirac matrices.
My first splitting/condensation is that I break the quarks and leptons into their "chiral" halves. That is, I assume that the electron is a composite particle composed of two parts, a left handed electron and a right handed electron. These two parts convert back and forth into each other with a probability that is proportional to the mass (more or less, neutrinos are a bit odd). The energy required to break, for example, an electron is approximately infinite (i.e. Plank mass) because the chiral states are massless and therefore travel at speed c.
This division of quark or lepton into two "constituents" is similar in that it is assuming that the quarks and leptons are composite particles that are "condensed" from simpler particles, but this layer of condensation is different from the ones following in two ways. First, the binding energy is "infinite", and second, there is a phase change here in that mass appears. The unbound particles travel at c while the bound particles travels have mass and travel at slower speeds. I like this way of going to the next preon state because it is compatible with the quarks and leptons being point particles (at energies smaller than the Plank mass), and it is the natural division of the quarks and leptons into subparts (i.e. the weak interactions treat the chiral particles very differently). This method treats mass as just another interaction between particles.
My second splitting/condensation is to break the chiral particles into three subparticles that I am now calling "snuarks". You can get my snuarks by taking Lee and Sundance's helons, and (a) splitting them into left and right halves, and (b) splitting H_0 into a particle / antiparticle pair. Consequently, I have eight snuarks while they have 3 helons, but as far as preon models go, the two models are fairly similar. For a while I thought that I could get an algebraic model that would underlie their theory, after splitting the H_0 into two particles, and after combining the right and left handed snuarks. The basic problem with getting my version to line up with theirs is that I have complex numbers (or things that act like complex numbers anyway), and I can't see how to get that in their theory.
My final splitting is to take the snuarks, and break each of them up into two particles that are assumed to be the truly elementary particles which I call "binons". In terms of the Geometric algebra, these are "primitive idempotents", which is what the mathematicians use when they want to say "\rho^2 = \rho and you can't make them any simpler". In terms of the Dirac matrices, the binons are the set of all possible density matrices, which is why I am constantly harping on density matrices on physics forums. In standard quantum mechanics, the spinor wave states are fundamental and the density matrices are derived from these. In my version of QM, these are reversed. My version is much more elegant, as you can see by reading the short discussion in
https://www.physicsforums.com/showthread.php?t=124904 but this simplicitly and elegance only happens with spin-1/2 density matrices, which is why I use them. This gives a geometric foundation for the elementary particles which is a great feature of my theory.
I call them "binons" because their quantum numbers of Clifford algebra primitive idempotents are binary (look up the "spectral decomposition theorem" for Clifford algebras to see a proof). As with any symmetry class, there are a bunch of different ways to choose good quantum numbers. You need 8 different types of binons to get the quarks and leptons. There quantum numbers are therefore (\pm 1, \pm 1, \pm 1), where the signs are chosen independently. Quantum numbers are additive. We can assume that the first of these quantum numbers depends on orientation, that is, it is the usual spin-1/2.
Binons are bound together by a potential energy. Since the binons are represented by Clifford algebra numbers, it is natural to use those numbers to define the potential energy. The definition is very simple. One adds together the Clifford algebra numbers, and then computes the "absolute value squared" of the sum. For the Dirac algebra, you could define the "absolute value squared" as the function which takes a matrix and gives the sum of the absolute squares of all its entries. That is a rather ugly definition (since it depends on the choice of representation etc.), but it turns out to be compatible (at least for the usual representations that physicists use) with the unique natural definition. The scale of the potential energy is the Plank mass. It is the potential energy that determines how binons have to be combined to make low energy particles and it is a great success of this theory that one can derive the structure of the quarks and leptons from first principles this way, with such a simple definition.
The snuarks are composed of two binons that are "compatible" in that their direction of travel is identical. In this, my theory is similar to the old "zitterbewegung" theory. The direction in which a chiral spin-1/2 particle travels is completely determined by its spin orientation, so the requirement that the snuark be compatible in their direction of travel is equivalent to requiring that their first quantum number be the same.
This leaves the other two quantum numbers arbitrary. I suppose that the next quantum number is "weak isospin". There are four cases: (+1+1, +1-1, -1+1, -1,-1). These four cases divide into a weak isospin doublet (+1+1, -1-1), and two weak isospin singlets (+1-1), (-1+1). The usual weak isospin quantum numbers are obtained by dividing these values, (2,-2,0,0) by 4. This simple derivation gives both the correct SU(2) symmetry and the correct pattern of representations, a great success of this idea.
The three snuarks that make up half of a quark or lepton are oriented in different directions. It turns out that this is necessary from the way that the potential energy is defined; otherwise there would be no bound states with energies less than Plank mass scale. So my version of preons has that when you successivley break an electron up into its components you first find 2 chiral states, then 6 snuarks, and finally 12 binons. That three snuarks are bound this way gives rise to an SU(3) symmetry, which is a great success of this theory.
As Clifford algebraic numbers, the snuarks can be multiplied by complex constants. That means that there are multiple ways of combining them together. Since this is a theory based on (pure) density matrices, one finds the number of ways that one can combine three snuarks together by solving the basic pure density matrix equation: \rho^2 = \rho. It is a great success of this theory that when you do this, you obtain three solutions, which correspond to the three generations of elementary particles.
The derivation in the previous paragraph requires solving equations which have 3x3 matrices. One of the side effects of this is that one obtains a new way of expressing the masses of the leptons. Having done this, one finds that there are remarkable patterns in the lepton masses. In particular, it appears that there are discrete symmetries that define the hierarchy of the generations, and also the hierarchy from the neutrinos to the charged leptons. This is a great success of the theory and is discussed here:
https://www.physicsforums.com/showthread.php?t=117787
Carl
I knew I needed to get into those exceptional algebras for some reason!
A good way to approach this beast is by seeking advice from one of its greatest hunters, Ed Witten. Back in Dec. 2003 Witten showed http://arxiv.org/abs/hep-th/0312171" that the perturbative expansion of \mathcal{N}=4 super Yang-Mills theory is equivalent to the D-instanton expansion of the topological B model with target space \mathbb{CP}^{3|4}. (selfAdjoint referred me to this paper originally)