I’ve just submitted a paper, Density Matrices and the Weak Quantum Numbers to Foundations of Physics. There are things about the paper that I didn’t include, things that I didn’t think were appropriate to a journal submission and I thought I’d talk about them here, and explain what the paper is talking about to a more general (but still math/physics) audience.
The paper is on the subject of the weak quantum numbers of the left and right handed elementary fermions and anti-fermions. Ignoring color and generation, there are 16 of these quantum objects. I provide a method of defining these quantum numbers by an idempotency equation, that is, by solving an equation of the form . Since pure density matrices satisfy this equation, the calculation is a density matrix calculation based on the permutation group on 3 elements.
The usual method of elementary particles is to assume that a symmetry relates the quantum states. In this calculation, the quantum states themselves are assumed to be composed of group elements of the symmtry. This can be done in density matrix formalism because density matrices can operate on themselves. Also of interest are what happens when different density matrices operate on each other. Particularly when the density matrices are chosen from the basis states of a complete set of mutually unbiased bases. But that’s another paper (mostly written).
Elementary particle physics, and quantum mechanics more generally, has been wedded to symmetry techniques almost since the founding of the subject. The elementary particles are described with symmetries and Einstein’s relativity is also a symmetry (space = time, more or less). In the case of the weak quantum numbers, the appropriate symmetry is U(1) x SU(2). One normally sees this written in reverse order, as SU(2) x U(1), but in the above paper I’ve listed the weak quantum numbers in the order (U(1), SU(2)), so it seems more appropriate to reverse them here.
Brief Description of Weak Isospin
The quantum numbers of a representation or “rep” of SU(2) can be zero, positive or negative, but they are always multiples of 1/2. With the weak quantum numbers, SU(2) provides the symmetry of weak isospin which is designated as . (By the way, the “3” refers to the 3rd Pauli spin matrix, , the one oriented in the z direction. This is because, in the state vector treatment of spin-1/2 spinors, it is customary to use the basis defined by spin up/down. In this basis, the vector (1,0) represents spin up while (0,1) represents spin down. And this choice is from the fact that the 3rd Pauli spin matrix is diagonal.)
Getting back to SU(2) weak isospin, only two representations are used. The first of these two representations is the SU(2) “singlet”. It has only one quantum state with quantum number 0 so for these particles, weak isospin . Of the 16 left or right handed elementary fermions or anti fermions, 8 are weak isospin singlets. And of these 8, four are particles and four are antiparticles. The four particles are evenly split between leptons and quarks and the four antiparticles are also evenly split.
The other rep is the SU(2) “doublet”. This rep has two states with quantum number +1/2 and -1/2. The 8 left or right handed fermions or anti fermions that are SU(2) weak isospin doublets are grouped into four pairs of two each. Two of these pairs are particles and the other two are antiparticles. The two particle pairs include a pair of leptons and a pair of quarks. Similarly for the antiparticles. Still restricting ourselves to the first generation, the SU(2) doublet of leptons consists of a neutrino and an electron, while the SU(2) doublet of quarks has an up quark and a down quark.
The weak force is carried by the W and Z bosons. Of the handed elementary fermions, only the SU(2) doublets feel the weak force. That is, only they can absorb or emit a W or Z. The weak isospin singlets do not interact with the weak force at all. The W is a charged particle with charge +1 or -1. In order for a fermion to emit one of these, the charge on the fermion must change. When this happens, the fermion converts over to the other weak isospin doublet partner. Since the SU(2) partners are pairs of leptons or quarks, this means that a neutrino changes to an electron (or vice versa, or with the antiparticle pair), and similarly a down quark changes to an up quark or vice versa.
Brief Description of Weak Hypercharge
The U(1), or circle symmetry group has representations for all integers N, positive or negative. Unlike SU(2), all U(1) representations are singlets. These representations are homeomorphisms of the circle group: . The integer N is a “winding number”, it tells you how many times you wind around the circle in the map before you get back to the start. So one would think that the available quantum numbers for weak hypercharge would be all integers. Uh, well, that’s not how it turned out…
The left handed leptons have weak hypercharge quantum numbers of -1, so their transformation law is . Similarly, the right handed electron has quantum number -2 which gives . (Or is it +2, I forget.) In these transformations, is the “charge“, which means “the generator of a continuous symmetry.” In this phrase, the word “generator” means that you can make small (continuous) changes in the value while keeping your equations still satisfied.
Electrons and protons have the same (but opposite) electric charge and anything made from them has integer electric charge. Naturally people assumed that this would apply to any elementary particle and this assumption got frozen into the definitions. As a result, when it was found that it took three down quarks to equal the electron’s charge, the value of for weak hypercharge ended up taking it on the chin. The quarks ended up with fractional weak hypercharge. If we multiplied all weak hypercharges by 3, they would be all integers, leptons and quarks. So if this bothers you, you can think of them this way, or perhaps write a physics paper to correct the injustice in a manner more complicated than to simply multiply all the weak hypercharge quantum numbers by 3. By the way, I should mention that there is already a factor of 2 between weak hypercharge and electric charge. This suggests that the true unit of electric charge is 1/6 the charge of the electron.
The course of elementary particle physics over the last 50 years has been to assume that the fundamental laws of physics are symmetry laws. One then makes the assumption that the symmetry laws are simple. After finding that a symmetry is only approximate, one uses the technique of broken symmetries. Basically, this idea is to suppose that the underlying differential equations are perfectly symmetric, but that their low temperature realization breaks the symmetry. Of course, as in epicycles of early astronomy, you can approximate any old thing by postulating complicated enough symmetry breaking.
The alternative technique I would prefer to use is to suppose that the universe is organized around differential equations and that these differential equations are simple. Symmetry principles are frequently useful in solving differential equations, but in my view, one should being so married to symmetry that one is unable to solve a problem any other way.
The Foundations of Physics
In the case of the weak quantum numbers, the foundational problem is that of explaining why they happen to be what they are. From the symmetry point of view, this is an unanswerable question. One might take the anthropic point of view and try to show that other universes would be incompatible with intelligent life, but this is a rather difficult task.
A few years ago, I independently realized that there was an alternative way of writing special relativity, a way so that it was no longer a principle of symmetry, but instead was a description of a perfectly realizable manifold with normal material properties (except that we cannot fashion things using an extra dimension). This form of special relativity is called Euclidean Relativity. My version amounts to assuming a single hidden, cyclic, very small, dimension. I assume that this is the origin of the U(1) symmetry. With this idea, all classical objects then move at speed c. If they are massive and are stationary, then their velocity in the hidden dimension is c.
The primary attraction of this idea for me was not that it eliminated symmetry, but instead that it would allow a definition of quantum mechanics where quantum waves would not be dispersive. A non dispersive wave travels at the same speed regardless of its frequency. In normal quantum mechanics, particles with different energies travel at different speeds and this requires dispersion. Dispersion is difficult from a mathematical point of view; having a non dispersive quantum wave theory means that calculations are easier. And in fact, the Wick rotation commonly used in quantum mechanics amounts to a transform to Euclidean relativistic coordinates.
It turns out that no one wants to hear about this sort of foundational ideas. It’s clear that one can move back and forth between standard relativity and Euclidean relativity so there is little motivation for physicists to look at ideas that come from Euclidean relativity. And most of the people working on Euclidean relativity ideas are amateurs. It smells of crankdom. However, if the underlying structure of the universe is, in fact, Euclidean instead of Einsteinian, then one is likely to find insights into the structure of the elementary particles by rewriting elementary partices from a Euclidean point of view.
If all you do is switch from Einstein’s relativity to Euclidean relativity you will accomplish nothing. To get anywhere you have to take the principle that you used to prefer Euclidean relativity (that of avoiding symmetry in favor of geometry), and apply it to all of elementary particles. Since elementary particles is built from symmetries, this is a tough roe to hoe. And I’ve got the calluses to prove it.
The other part of the title of the paper, “density matrices” describes how the calculation is set up. Density matrices are an alternative formulation of quantum mechanics that is not as commonly use as the usual state vector formulation. The advantage of density matrices is that they do not possess the arbitrary U(1) symmetry (which is more or less related to weak hypercharge) of the state vector formulation. Since my objective has been to rewrite elementary particle physics without recourse to using symmetry principles in the foundation, it is natural for me to prefer the density matrix formulation.
So finding the equations that define the weak quantum numbers was not a difficult task. This was done after I had used a very similar method to rewrite Koide’s wave equation as an eigenvalue equation and extended it from the charged leptons to the neutrinos. Knowing that the elementary fermions are composed of three identical preons, I was sure that the appropriate group would be the symmetry group on 3 elements. From the permutation group on 3 elements, it takes only a few minutes to write down the 6 coupled quadratic equations. Actually solving them took me 2 or 3 days of hard work. When you’re as slow as I am at calculations it helps to be able to restrict the problems you work on to the ones that will give useful results.
The calculation as submitted is not at all how I thought of it when I first began working on it. To me, the six elements I, J, K, R, G, and B are analogs of Feynman diagrams. Each represents a quantum amplitude. The requirement of idempotency is equivalent to requiring that the particle be in a stable state. This is all more completely described in the paper I wrote up when I had first made the calculation Density Operators, Spinors and the Particle Generations. Like the just submitted paper, this was intended for Foundations of Physics. But the length basically grew to infinity. The new calculation is improved in that it uses the principle of Hermiticity to separate the extra solutions of the idempotency equation from the 16 that represent the 16 handed elementary fermions or anti fermions.