In the comments to my previous post, Tony Smith asked where color came from in my use of the Clifford algebra C(4,1) as I didn’t explain it in my previous post. This is one of the 30 or so topics on which my guesses for the physics of sub elementary particles differs drastically from mainstream physics.
Crack(pot)s in the Foundations
The problem with making drastic changes to the foundations of physics is that the foundations are tightly woven together with very long threads. When you pull a thread out, you find that there is a neat whole left which just happens to be exactly the size and shape of the thread you pulled out. When you try to weave a new thread in a new direction starting in part of the hole left by the one you yanked out, you find that there are many other threads that get in the way. You have to pull those threads out too. And then these changes cascade to yet more changes.
By the time you are done, you will find that you have to rewrite the foundations completely. This is why people who mess with the foundations of physics are thought of as crackpots; they almost always are.
The Pauli exclusion principle says that there are only two linearly independent states for a spin-1/2 qubit. One can choose whatever axis one wants to split the states, z is traditional.
If we happen to have a system that is naturally modeled by qubits because it happens to have no spatial dependency (as happens when condensed matter physicists make quantum dots), then it would be perfectly understandable, in a classical sense, that there are only two degrees of freedom in the lowest energy state. The lowest energy spherical harmonic is a singlet. And electrons come in two versions, spin up and spin down, thus we have 2 linearly independent low energy states.
This is just what we would expect for the classical thermodynamics of waves. The classical thermodynamics of particles is unsuitable for elementary particles because elementary particles are indistinguishable. But classical waves are indistinguishable so their statistics match that of quantum mechanics, at least when we look at just one qubit. To get them quantized, you assume that what we call “particles” are just solutions to the differential equation that persist, i.e. solitons.
It’s often said that classical statistical mechanics uses Maxwell-Boltzmann statistics, but this is true only for classical objects that are distinguishable. Solitons, as one might model in a perfectly classical (nonlinear) differential equation, are indistinguishable in that when you transform the differential equation into a system of creation and annihilation operators, the operators commute. But one cannot put two solitons at the same point because the differential equation they satisfy has to be nonlinear. This is a sort of Pauli principle.
Massless Handed States and Spherical Harmonics
The precolor model I play with is not a model of qubits in the usual condensed-matter fashion. The structure of the standard model suggests that the massless handed states, like the left-handed electron, should be more elementary than the elementary particles. In this model mass is just a coupling between them the two handed particles. Since the handed states are massless, they travel at speed c. Because of this, they cannot be treated as a singlet in spherical harmonics. To have a velocity vector, they need to be a vector state. In the spherical harmonics, this makes them a triplet.
Let me repeat this argument. So long as you think of the electron as the fundamental particle, the classical statistical mechanics of waves will give you the correct statistics for the number of linearly independent states. But when you follow the lead of elementary particles, and suppose the left and right handed states to be the fundamental states, then classical statistical mechanics of waves does not work. The waves have to travel at speed c, and by the assumption that the universe has rotational symmetry and is described by a differential equation, separation of variables gives spherical harmonics. The singlet states are spherically symmetric and therefore do not have a direction in which they can travel. The next lowest energy states are triplets, and they do have orientation.
So in short, precolor comes from making the assumption that the preons count linear independence in the manner that classical mechanics would count soliton traveling waves. Following the chemical notation, the spherical harmonic triplet 2Px, 2Py, 2Pz waves becomes the red, green and blue colors.
The Coleman-Mandula Theorem
This assumption is what you would expect if you wanted to describe quantum physics of elementary particles in a classical manner. Of course there are a bunch of places where this contradicts the standard version of physics. I believe I’m aware of most of them, the number is around 30. In the rest of this post, I’ll talk about 3 of them, the Coleman-Mandula theorem, the speed of preons, special relativity violations, and then talk about how these sorts of things show up when you mess around in quantum gravity.
A first objection is that this relates color to orientation in the same way that spin is related to orientation. There is a “no-go” theorem that says (more or less) that this cannot be done, the Coleman-Mandula theorem. Ignoring some reasonable technical assumptions, the Coleman-Mandula theorem says that other than spin, all conserved quantities have to be Lorentz scalars. Another way of saying this is that the SU(3)xSU(2)xU(1) of the standard model cannot have anything to do with the SU(2) that arises from special relativity.
The Coleman-Mandula theorem must have been a very depressing piece of work to people who wanted to describe the elementary particles from a geometric point of view, particularly those who would like to expand special relativity. Of the threads that support C-M, the one that has to be yanked in my version of the elementary particles is the assumption of perfect Poincare symmetry.
Another problem with supposing that precolor is the triplet of spherical harmonics is that it requires that a handed elementary particle be built from waves that travel in three different (perpendicular) directions, for example, in the +x, +y, and +z directions. The overall particle would then move in the (1,1,1)/sqrt(3) direction. But if a soliton wave is traveling at speed c in the +x direction, its component of velocity in the (1,1,1)/sqrt(3) direction is only c/sqrt(3). This last problem bothered me for quite some time. Then I realized that one could get the particle to travel at speed c again if you assumed that the three preon / solitons traveled at speed c sqrt(3).
The special theory of relativity says that stuff moving around at speed c sqrt(3) is impossible. But the failure to observe preons can be attributed to their having energies the order of the Planck mass. And what’s more, the Coleman-Mandula theorem assumes perfect Poincare symmetry. So both these problems can be eliminated by assuming that special relativity is just an “effective theory”, a low energy approximation a fate similar to Newton’s gravity but with low energy rather than low velocity being the approximation regime.
Historically, the assumption has been that quantum mechanics supplanted classical mechanics so using analogies based on classical soliton waves aren’t useful. I think that these sorts of conclusions shouldn’t be jumped to, at least until we have a unified theory of gravity and quantum mechanics. And there are some interesting papers on quantum gravity that suggest that quantum mechanics is inconsistent by a factor of three.
A fascinating paper by Shahar Hod is “Bohr’s Correspondence Principle and the Area Spectrum of Quantum Black Holes”, Phys. Rev. Lett. 81, 4293 (1998), gr-qc/9812002. The paper uses Bohr’s correspondence principle to show that the areas of black holes are spaced by the interesting factor of . The ln(3) is unusual here.
A few years later, Lubos Motl used Schroedinger’s equation to make the exact calculation: “An analytical computation of asymptotic Schwarzschild quasinormal frequencies”, Adv. Theor. Math. Phys. 6 (2002) 1135-1162, or gr-qc/0212096, is about the thermal physics of black holes and quantum gravity. This verified the correctness of the formula deduced by Hod.
In his paper, Motl considers small perturbations to a black hole using Schroedinger’s equation. He computes the transmission amplitude as a function of frequency for various choices of spin j. There turned out to be three cases, Odd integer spin, even integer spin, and half integer spin. These apply to photons, gravitons and higgs, and the fundamental fermions like the electron. The transmission amplitudes, , are:
where is the Hawking temperature of the black hole.
Now the spin 1/2 and spin 1 particles are exactly what we would expect for the average occupation numbers for Fermi-Dirac and Bose-Einstein statistics. On the other hand, the even spin particles, which as far as elementary particles go would be the Higgs and graviton, break the Fermi-Dirac / Bose-Einstein statistics pattern. As Motl wrote:
This agreement makes the result (51) for the even values of j even more puzzling. Why do we fail to obtain the same Bose-Einstein factor as we did for odd j? Instead, we calculated a result more similar to the half-integer case, i.e. Fermi-Dirac statistics with the number 3 replacing the usual number 1; let us call it Tripled Pauli statistics. Such an occupation number (51) can be derived for objects that satisfy Pauli’s principle, but if such an object does appear (only one of them can bre present in a given state), it can appear in three different forms. Does it mean that scalar quanta and gravitons near the black hole become (or interact with) J = 1 links (triplets) in a spin network that happens to follow the Pauli’s principle?
It’s hard to see why gravitons “near the black hole” should be any different from gravitons far from the black hole. And this suggests that if my model of the preons is correct, gravitons need to treated as objects that travel at preon speeds, that is, faster than light by a factor of sqrt(3). And that is the theory of gravity I’ve been working on. It is rather fortunate that the speed of gravity has not yet been measured and there is at least hints of new evidence that it is not c.
In short, precolor is simply the classical spherical harmonic triplet degrees of freedom that are natural for a vector quantum state. I’ve left many many unfinished threads dangling, but this post is already long enough. A very obvious one is “how can it be that SU(3) color can be treated as an internal symmetry in the standard model, when in this model it arises as the remnant of an external force? Isn’t this incompatible with the usual SU(3) calculations?”