Last week’s announcement that the Nobel Prize
in physics had gone to the experimenters who first observed neutrino
oscillations reminded me that I hadn’t covered the relevant physics in this
blog. So here we go.

I told you what neutrinos are here,
and I covered quantum superpositions of spacetime points here.
In quantum mechanics a superposition of states is also a state, so
superpositions of points are also points. Particles are resonances of points. In
the latter post I showed how protons and neutrons and other baryons are resonances
of superpositions of three points. Mesons are resonances of superpositions of
two points. Electrons are resonances of single points. Thus, spacetime is
really a superposition of spacetimes for different values of n, where every
point in a particular spacetime is a superposition of n single or pure points.
All of the particles observed so far are in the n = 1, 2, or 3 spacetimes,
although there has been a recent report of a possible n = 5 sighting.

So far I haven’t put any restrictions on the
positions of the points in a superposition, except that they have to be close
enough to each other that when a resonance moves from one to another , nothing
exceeds the speed of light. This gives us all of the hadrons—mesons and
baryons. It also gives us the concept of quarks. Experimentally, these
composite particles look like combinations of smaller particles, which are
called quarks. They’re really just parts of a resonating superposition of
points, not separate particles, but the concept has proved useful. There are
three generations of quarks, one for each value of n.

Now I have to tell you about the
leptons—electrons, muons, and taus. Electrons you’ve met. Muons and taus appear
to be single-point particles like the electron, but heavier. Actually, they’re
resonances of n =2 and n = 3 superpositions, but with a tighter restriction on
the positions of the constituent points. The points must be adjacent and rotate
through a common position, like a little eddy in space. Each point resonates
only when it’s in the common position, so the resulting particle doesn’t seem
to have parts, but it actually does have parts, two for muons and three for
taus.

So now we have three generations of leptons,
that is, the electron, muon, and tau particles, along with the corresponding
neutrinos—the same resonances moving at the speed of light. We have seen how
the first-generation quarks come to exist. The second and third quark
generations are formed by applying the same principles to the muon and tau
spacetime point patterns. Because the muon and tau patterns look like electron
patterns from the outside, they can form superpositions of states to generate
other hadrons, resulting in the second and third generations of quarks.

Neutrino Oscillations

When we discussed neutrinos (here

__)__, we saw that the neutrino can be seen as a particle whose speed oscillates sinusoidally around the speed of light. This is a classical picture that results from our choice of a gauge for our particle model. These are not the neutrino oscillations for which the 2015 Nobel Prize was awarded. The prize was for the discovery that electron, muon, and tau neutrinos can change identities in flight, turning from one type or flavor to another. We’ve said that electron, muon, and tau neutrinos are the corresponding lepton resonances moving at the speed of light. This sounds simple, but the reality is more complicated. Let's say an electron neutrino is created in a weak interaction. It is observed as a fermionic spacetime resonance moving at the speed of light. Is it a single point moving at the speed of light and resonating, or does the resonance move from point to point? It's impossible to say, so we must assume that both cases are possible. If the resonance moves from point to point, how can it be guaranteed that all of the points it touches are in the n = 1 spacetime? Recall that spacetime or the vacuum is a mixture of different vacua parameterized by an integer n. It is the n = 2 spacetime that gives us muons, and it is the n = 3 spacetime that gives us tau particles.
Nothing guarantees that an electron neutrino
resonance will always occur on an n = 1 point, so we have to conclude that an
electron neutrino must be a mixed state, that is, a resonance of a mixture of n
= 1, n = 2, and n = 3 points, because spacetime itself is such a mixture. The
ideal pictures of the electron, muon, and tau neutrinos, that is, resonances of
n = 1, n = 2, and n = 3 points, respectively, moving at the speed of light,
represent eigenstates of the system. However, these eigenstates are not the
same as the electron, muon, and tau neutrino eigenstates. The electron, muon,
and tau neutrino eigenstates are flavor eigenstates, while the n = 1, n = 2,
and n = 3 eigenstates are mass eigenstates. The mass increases with n because
of binding energy.

So here is the situation. For neutrinos, the
flavor eigenstates are not the same as the mass eigenstates, and the flavor
eigenstates are mixtures of the mass eigenstates (and vice versa, of course).
This, with the mass differences, meets the conditions for neutrino oscillations
in the most popular model. I won't go into the mathematics here because the
current literature presents it very well. Simply put, when an electron
neutrino, for example, is observed, it can look like another flavor because
it’s a mixture of all three mass eigenstates. The leaders of the Super
Kamiokande and Sudbury collaborations received the Nobel for actually observing
neutrinos changing flavor in flight.

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