I want to continue developing our spacetime model in some of its more sophisticated aspects. We need to know about left- and right-handed particles and why there are no right-handed neutrinos. We’ll show that when you have the correct spacetime model, these esoteric subjects aren’t as intimidating as they’re sometimes made to seem.
We’ll consider only the fermionic points for now. As we saw here, they are pulled together by gravity, but because they are fermions, they can’t all have the same position, so they are held apart by degeneracy pressure. At the end of inflation, they are as close together as the degeneracy pressure will allow. Each point (actually each point quad, see here) is trapped in a more or less spherical cell whose walls are other fermionic points. Within their cells, the points’ positions vary randomly from one time tick to the next. This vibrational energy has a ground state, approximately the Planck energy, 1019 GeV, but because each point vibrates independently of the others, over any significant volume of space this energy averages to near zero and we don’t see it. Some of the points are stationary, remaining in the same point cell from one time tick to the next, but most move to an adjacent point cell between time ticks. At any time, there are 1017 moving points for every stationary point. This number is determined by the Higgs field.
We’ve said that the points’ positions vary randomly from time tick to time tick, but what about the time between ticks? Well, according to Einstein’s theory of special relativity, distance and time are interchangeable, so spacetime looks the same whether we treat both position and time as random or ascribe all of the randomness to position and treat time as regular. Since it doesn’t matter, we elect to treat time as regular.
In addition to its ground state, each point can be in excited states with energy above the ground state. The first excited state, with twice the ground state energy, is stable. We see the energy above the ground state as a particle. We see excited stationary points as electrons and excited moving points as neutrinos. By the way, this doesn’t mean that electrons never move. A moving electron is some quantum superposition of a stationary excited point and a speed-of -light moving excited point, while a neutrino is 100% a moving excited point.
Now, just as we had a choice of where to put the randomness for unexcited points, we have a similar choice for particles. We can treat both the ground state energy and the particle energy as random position vibrations, or we can just treat the ground state energy as random and treat the particle part of the energy as some function of time. In our model, the local creation time of a point doesn’t have to be in step with the observer’s time or what we can call the global time. Special relativity guarantees that the excited point looks the same to an observer as long as the total energy stays the same. Therefore, we elect to treat the particle energy as a harmonic oscillator in which the local time at the excited point varies sinusoidally around the global time with frequency ν (Greek nu), so that the energy is hν, where h is Planck’s constant. This works for both electrons and neutrinos. However, while electrons usually don’t move at anywhere near the speed of light and don’t cause any trouble, a funny thing occurs with neutrinos.
A neutrino is a resonance of a fermionic point moving at the speed of light. It moves one point cell for each time tick, but its creation time oscillates around the global time. In other words, its instantaneous velocity oscillates above and below the speed of light, c. Now imagine that you are riding on a photon traveling parallel to a neutrino. When the neutrino’s speed drops below your speed, c, you see it behind you, spinning to the left. When the neutrino speeds up, it passes you and you see it ahead of you, spinning to the right!. So is it lefthanded or righthanded? Actually, it is both. When its velocity is less than c it is lefthanded and seems to have mass. When its velocity exceeds c it is righthanded and has imaginary mass. On average, neutrinos have zero mass and travel at the speed of light, but half the time they look like massive particles and half the time they are tachyons, sterile and undetectable.
That brings us to an esoteric little concept called weak isotopic spin, or weak isospin, or just isospin. It’s not hard tp see that electrons and neutrinos are really the same particle, except that one moves at the speed of light and the other is what we consider stationary, at least when compared to the speed of light. Physicists say that electrons and neutrinos are a weak isospin doublet. They morph into each other by exchanging W and Z bosons in a process known as the weak interaction or weak force. When one changes to the other it is said to undergo a weak isospin rotation or an SU(2) rotation, where SU(2) is the special unitary group of transformations with two elements. What’s actually rotating here? We are dealing with four-dimensional spacetime, but each point has its own local four-dimensional version of that spacetime, and it doesn’t have to line up with the observer’s or global version. An isospin rotation is really just a rotation of the local time axis with respect to the global time axis. When the local time axis lines up with the global time axis, we have a stationary particle, an electron. When the local time axis lines up with zero global time, we have a speed-of-light particle, a neutrino. Remember that the theory of special relativity says that to a stationary observer, time seems to stop for an object moving at the speed of light. The W and Z bosons just take care of the bookkeeping needed to do the standard model calculations.
The “spin” in the name isospin got there because physicists knew the electron-neutrino symmetry signaled the existence of a new quantum number that had to be intrinsic, like spin. They still don’t know what’s really going on, and they have no idea that position and time are also intrinsic to spacetime points. They don’t even know that points exist.
Now back to handedness. There are both lefthanded and righthanded electrons. Apply an isospin rotation to a lefthanded electron and you get a lefthanded neutrino. Apply an isospin rotation to a righthanded electron and you get…nothing. Neutrinos aren’t detectable when they’re righthanded. Conclusions? There are no righthanded neutrinos, and righthanded electrons are an isospin singlet (i.e., they have zero isospin).
OK, enough for now. I hope I’ve shown that even some concepts—like isospin—that are puzzling to physicists are really pretty straightforward when you know what’s going on.