The standard theoretical model is to assume that there are three different neutrinos with different rest masses in the range 0.05 eV ... 0.5 eV: ν₁, ν₂, ν₃.
Let us have a reaction which creates an electron neutrino νe. The neutrino can be any of mass states ν₁, ν₂, ν₃. We do not know which it is. Various rest masses have various probability amplitudes.
Let α = e. The probability of observing the neutrino as the flavor β at the distance L from its creation is:
where mj are the various neutrino rest masses, and Uαj are constant components from the Pontecorvo et al. 3 × 3 matrix.
The formula on the right sums the waves of different particles: neutrinos with different rest masses m₁, m₂, m₃? That is strange. Is that allowed in quantum mechanics?
In quantum electrodynamics, the electron Dirac field is classical, and can describe an electron or a positron. There we have another case in which we sum waves of different particles: the electron and the positron.
Suppose that a neutrino with a rest mass m flies in vacuum. Obviously, the rest mass of the neutrino cannot change. There cannot be an oscillation of neutrinos if we look at their rest mass.
Anca Tureanu (2025) says that created neutrinos form a "statistical ensemble" and we cannot use the interference of their wave functions
Professor Anca Tureanu from the University of Helsinki says that, in quantum field theory, one is not allowed to calculate interference for wave functions of different particles born in a reaction.
A principle in quantum field theory, and in quantum mechanics, is that we can only calculate an interference of two waves if we cannot know which of the two wave histories happened. In the double-slit experiment, we cannot know if the photon passed the left or the right slit. We are allowed to calculate the interference pattern of the two different paths of the photon. But if we place a detector which can determine the path of the photon through the slits, the interference pattern disappears.
If we know the location of the reaction well, we do not know the mass state of the created neutrino
Suppose that we prepare the incoming particles in such a way that their momentum (and energy) are known extremely precisely. We measure the energies and the momenta of outgoing particles, except of the neutrino. That way we will know the energy and momentum of the created neutrino, and can determine its rest mass m.
The uncertainty principle says that we then cannot know the location of the reaction precisely:
Δp Δx ≥ ħ / 2.
The flavor probability formula has the parameter L which tells the distance to the reaction.
The energy-momentum relation is
E² = p² + m².
Let E and |p| be roughly 1, and m ≈ 10⁻⁶. To determine a rough value of m from the energy-momentum relation, we have to know p to the precision better than Δp = 1/2 * 10⁻¹². We use natural units, so that ħ = 1. Then
Δx > 10¹².
We see that the formula
exp( -i mj² L / (2 E) )
in the flavor probability equation above then has a relatively large uncertainty because of L:
(10⁻⁶)² * 10¹² / 2 = 0.5.
That is, we know the rest mass of the neutrino, but do not know the neutrino oscillation phase too well. The uncertainty, actually, is quite a lot larger than 0.5 because we must measure momenta of several particles. Also, we chose Δp too optimistically above. A better choice might be 1/3 of the value we chose. Then the uncertainty 0.5 easily becomes 3 or more.
That is, we cannot know the phase of the neutrino oscillation. The flavor probability formula in this case gives the average over a whole cycle of the oscillation. The average is a constant and does not depend on the value of L. This is reasonable.
If we know L relatively precisely, then we cannot know the rest mass state of the neutrino. It makes sense to calculate the sum (interference) of different rest mass waves.
What is the created neutrino? Which state does it have?
If we know the location of the reaction relatively well, then we cannot know the rest mass of the created neutrino.
The created neutrino can then be seen as a "malformed" classical neutrino field wave, which does not have a definite rest mass until we measure the rest mass in some way.
There is only one neutrino particle which has 3 different states? In the model above, there is a single neutrino field which has (stable?) states of three different rest masses. Should we say that the neutrino is just one particle, which appears in three stable states?
An analogous question: are the electron and the positron a single particle which has two states?
Can a neutrino decay to a lower rest mass state? Could it emit a photon?
*** WORK IN PROGRESS ***
