I’m releasing two papers that relate Heisenberg’s uncertainty principle, spin-1/2, the generations of elementary fermions, their masses and mixing matrices, and their weak quantum numbers. I haven’t blogged anything about these because I’ve been so busy writing, but I should give a quick introduction to them.

Heisenberg’s uncertainty principle states that certain pairs of physical observables (i.e. things that physicists can measure) cannot both be known exactly. The usual example is position and momentum. If you measure position accurately, then, by the uncertainty principle, the momentum will go all to Hell. That means that if you measure the position again, you’re likely to get a totally different result. Spin (or angular momentum), on the other hand, acts completely differently. If you measure the spin of a particle twice, you’re guaranteed that the second measurement will be the same as the first. It takes some time to learn quantum mechanics and by the time you know enough of it to question why spin and position act so differently you’ve become accustomed to these differences and it doesn’t bother you very much.

If you want to figure out where an electron goes between two consecutive measurements the modern method is to use Feynman’s path integrals. The idea is to consider all possible paths the particle could take to get from point A to point B. The amplitude for the particle is obtained by computing amplitudes for each of those paths and adding them up. The mathematical details are difficult and are typically the subject of first year graduate classes in physics. Spin, on the other hand, couldn’t be simpler. Spin-1/2 amounts to the simplest possible case for a quantum system that exhibits something like angular momentum.

The simplest possible case for something like Heisenberg’s uncertainty principle occurs in spin-1/2. This is a 2-dimensional Hilbert space so calculations are a lot simpler than the infinite dimensional case needed for position and momentum. In these finite dimensional Hilbert spaces the quantum information theorists refer to two observables as “mutually unbiased” if they are related the way that position and momentum are. That is, knowing one of the observables exactly means that you can know nothing at all about the other observable. In terms of quantum mechanics that means that the transition probabilities are all equal, or unbiased.

A paper by George Svetlichny describes Feynman’s path integrals using the concept of MUBs:

Feynman’s Integral is About Mutually Unbiased Bases, George Svetlichny, 2007:

*The Feynman [position path] integral can be seen as an attempt to relate, under certain circumstances, the quantum-information-theoretic separateness of mutually unbiased bases to causal proximity of the measuring processes.*

My two latest papers apply the idea of path integrals being about mutually unbiased bases to spin-1/2. Under this assumption, spin is no longer an observable that stays constant after you measure it. Instead, it jumps around just like position. So the papers are about applying the idea behind Heisenberg’s uncertainty principle to the simplest quantum mechanical system possible, the qubit (or spin-1/2). The two papers are for Foundations of Physics. The first is:

Spin Path Integrals and Generations, Carl Brannen, 2009:

*Two consecutive measurements of the position of a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have explained this property of position observables as a result of a path in the Feynman integral being mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). On the other hand, two consecutive measurements of a particle’s spin give identical results. This raises the question “what happens when spin path integrals are computed over products of MUBs?” We show that the usual Pauli spin is obtained in the long time limit along with three generations of particles. We propose applications to the masses and mixing matrices of the elementary fermions.
*

The above paper is in the “under review” stage at Foundations of Physics. This seems to be the middle stage for a paper, the first stage is “editor assigned” or something like that. The paper buries deeply into the shaky foundations of physics. Those who are working on the periphery of physics imagine that the foundations are more stable than they are and will find the paper a bit shocking. For laughs, I’ve linked in the copy that I uploaded to arXiv. ~~Some moronic Cornell grad student moderated it off.~~ Maybe I’ll resubmit it when the paper is accepted for publishing, maybe not. The second paper is:

Path Integrals and the Weak Force, Carl Brannen, 2009:

*In a previous paper, we showed that spin-1/2 can arise from a more primitive form of spin called “tripled Pauli spin”, along with three generations of elementary fermions. This gave a possible explanation for various coincidences in the fermion masses and mixing matrices. In this paper we continue the analysis. We show that the weak hypercharge (t_0), and weak isospin (t_3) quantum numbers can be derived from the long term propagators of three tripled Pauli spin particles. This completes a derivation of the standard model elementary fermions.*

The second paper should be ready to submit to Foundations of Physics around the end of the month. The above is a first draft. Of course I’d appreciate comments and corrections. Right now I’m thinking that my next paper will be about what all this says about the geometry of spacetime. Or maybe I’ll write an introduction to Clifford algebra.

Hi Carl,

I wonder if you’ve thought about whether the square roots of mass in the Koide formula are linked to the Weyl 2-spinor (left and right handed spinors) using Schroedinger’s ‘Zitterbewegung’ lepton as discussed by Penrose, Road to Reality, 2004. See Penrose’s Figure 25.13 for a Feynman diagram showing how two components of a lepton interact to produce the de Broglie oscillation of a moving particle. They acquire mass at the interaction vertex. The Zitterbewegung vertex coupling constant for the interaction is a square root factor, because you square the momentum integrated amplitude of coupling constants and propagators for a Feynman diagram to get the resultant probability or reaction rate.

Because Zitterbewegung involves interactions between two components, a zig and a zag, in a lepton, you need two interaction vertices in each complete oscillatory cycle of the particle as it propagates, and the coupling constants multiply together to give the amplitude according to Feynman’s rules; this provides the seed for an explanation of the square roots in the Koide formula. Since any lepton is acquiring mass from the same vacuum at vertices, it follows that to explain the variety of lepton masses that exist in nature, the Zitterbewegung vertex coupling constants must be proportional to the square root of the mass of the zig and zag components of a Zitterbewegung lepton. We know that neutrinos oscillate in flavour uniformly between three flavours while coming to us from the sun, which is why we detect just one third of the total (the third which have the flavour that our discriminate detector is searching for). If we extend this idea so that the zig and zag components of leptons in general, the masses of leptons will be represented by the sum of products of pairs of square roots of masses of different leptons. It’s possible to rewrite the Koide formula such that the sum of the 3 lepton masses (electron, muon and tauon) is equal to 4 times the sum of the 3 products of the square roots of those 3 different masses. I’m want to know if this links up with your formula for neutrino masses. (I haven’t investigated beyond this yet, but if you’ve already looked at this approach please let me know so I don’t waste time going over old ground.)

Nige, I’ve been wondering if the square root comes from the left and right halves being treated differently. Say the mass interaction is defined only by the left handed fermions. Then you get natural square roots. Do keep working, it may be a waste, it might even be on old ground, but it’s worth thinking about.

All, I should admit that I’m going to have to rewrite the second half of this paper substantially because there’s an arithmetic error in the calculation of the solution to the four coupled quadratic equations. In short, I left some solutions out. If you’re looking for entertainment, try solving the four coupled quadratic equations in (14), subject to , and see if you get (15) and (16), or if you instead get a little extra.

I’ll explore some ideas for making the unwanted solutions go away tonight and fix it soon.

“…a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny…”

should be “non-relativistic”. Unfortunately, I’m not nearly qualified to check the math 😦

Good luck 🙂