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In summary, there is an assumption that if neutrinos didn't have mass, they would move at the speed of light due to the E^2 -p^2 = m^2 equation. However, the fact that they oscillate proves that they do have mass. The oscillation phase is directly related to the difference in mass between the three types of neutrinos, meaning that at least one of the masses must be non-zero. The concept of oscillations in neutrinos is similar to the precession of a spin in a magnetic field and is also seen in the quark sector.

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Because of the [itex]E^2 -p^2 = m^2[/itex]Kiley said:Why is there an assumption that if neutrinos didn't have mass they would move at the speed of light

When the mass is 0, then the particle moves at the speed of light and there is no reference frame where it appears at rest.

Because the oscillation pattern appears with the mass-squared difference between the neutrinos... if they are massless (or even mass-degenerate) there wouldn't be any oscillation visible.Kiley said:and how does the fact they oscillate prove they have mass?

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The oscillation phase in neutrino oscillations is directly proportional to the difference between the squares of the masses of the different neutrino mass eigenstates. This means that at most one out of three neutrino masses can be zero.

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Any massless particle is required to travel at c in a vacuum. Massless neutrinos would travel at c for the same reason massless photons do.Kiley said:

The oscillations occur because the the wave length of three mass states (electron, muon, and tau) are different, and thus shift in phase with respect to each other. This phase shift is what creates the oscillation. But for the wavelengths to be different, the mass of the three types of neutrinos have to be different, which in turn means they must be finite and non-zero.

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Thank you all for your replies. Could you recommend any literature of the subject?

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http://web.mit.edu/shawest/Public/8.06/termPaperDraft.pdf

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Thank you so much Jtbell, this is very helpful! :)

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Some small bit of nit-picking. The electron muon and tau states are not the neutrino mass states. In fact, it is crucial for oscillations that not only the masses of the mass eigenstates are different, but also that the flavour states, i.e., the electron muon and tau neutrino states, areJanus said:three mass states (electron, muon, and tau) are different

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The mass eigenstates are the HamiltonIan eigenstates,but what about the flavour eigenstates,which operator's eigenstates they are?Orodruin said:Some small bit of nit-picking. The electron muon and tau states are not the neutrino mass states. In fact, it is crucial for oscillations that not only the masses of the mass eigenstates are different, but also that the flavour states, i.e., the electron muon and tau neutrino states, arenotequivalent to the mass states but instead are linear combinations of the mass states.

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Orodruin said:Some small bit of nit-picking. The electron muon and tau states are not the neutrino mass states. In fact, it is crucial for oscillations that not only the masses of the mass eigenstates are different, but also that the flavour states, i.e., the electron muon and tau neutrino states, arenotequivalent to the mass states but instead are linear combinations of the mass states.

The above has confused me for a while, are there any other quantum phenomena in nature that have similar physics to the above?

Thanks!

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Quarks. The difference in the quark sector is that the masses are so different that any interference between the mass eigenstates quickly is subject to decoherence. This is why you have W interactions changing the quark generations.Spinnor said:The above has confused me for a while, are there any other quantum phenomena in nature that have similar physics to the above?

Thanks!

In terms of quantum oscillations, the mathematics is completely analogous to having a spin precessing in a magnetic field that is not parallel to the direction you are measuring the spin component in. For example, you can have a spin-1/2 particle and measure its x-component to be positive at t = 0. Applying a magnetic field in the z-direction, the probability to measure a positive x-component will oscillate between 0 and 1.

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Can I ask a question about the mass eigenstates and flavor eigenstates?Orodruin said:Quarks. The difference in the quark sector is that the masses are so different that any interference between the mass eigenstates quickly is subject to decoherence. This is why you have W interactions changing the quark generations.

In terms of quantum oscillations, the mathematics is completely analogous to having a spin precessing in a magnetic field that is not parallel to the direction you are measuring the spin component in. For example, you can have a spin-1/2 particle and measure its x-component to be positive at t = 0. Applying a magnetic field in the z-direction, the probability to measure a positive x-component will oscillate between 0 and 1.

the thing that I understand is that mass eigenstates do not oscillate but the flavors do,and the flavor eigenstates are the superposition of mass eigenstates,and they relate together with a unitary transformation matrix called PMNS,and still we do not know the value of few of neutrino parameters in this matrix, if we find the value of these parametes do we can find the exact value of the neutrinos,like the mass of the electron neutrino or muon neutrinos?

indeed I think I misunderstood the qunatum interpretation of this phenomena,can you help me to understand it?

thanks.

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There is no such thing.kimmm said:like the mass of the electron neutrino or muon neutrinos

The electron neutrino is a superposition of the three mass eigenstates. It doesn't have a well-defined mass. It only has an expectation value for the mass.

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thanks for the reply but still I do not understand the quantum phenomena,so how the superposition is defined by PMNS matrix,and if we know the values of the matrix elements we can not find the definite masses, I mean by mathematics, can you help me to understand how a particle can not have a definite mass?mfb said:There is no such thing.

The electron neutrino is a superposition of the three mass eigenstates. It doesn't have a well-defined mass. It only has an expectation value for the mass.

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We can. Those are the masses of the mass eigenstates. General linear combinations of those do not have a definite mass.kimmm said:and if we know the values of the matrix elements we can not find the definite masses

It is no different from whatever quantum system. You can have energy eigenstates that have a definite energy and you can have linear combinations of energy eigenstates that, generally, do not.

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So I did a Google image search, "the mathematics of neutrino oscillation" and came up with the following image,

What more, if anything, than the above is needed to understand neutrino oscillation physics?

Edit, the above just looks like a change of basis?

This is all of it?

Thanks!

What more, if anything, than the above is needed to understand neutrino oscillation physics?

Edit, the above just looks like a change of basis?

This is all of it?

Thanks!

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Orodruin said:We can. Those are the masses of the mass eigenstates. General linear combinations of those do not have a definite mass.

So we will never know the exact masses of neutrinos(flavor ones) , as they define by superposition of states?

but just the mass eigenstates by the improvements of experiments,the normal hierarchy or inverted I mean.

is that right?

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It is not a question of not knowing the masses of the flavour eigenstates, it is a matter of the flavour eigenstates not having definite masses.kimmm said:So we will never know the exact masses of neutrinos(flavor ones) , as they define by superposition of states?

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That depends on what you put into the word "understand". Effectively, it is just the evolution of a three-state quantum system.Spinnor said:What more, if anything, than the above is needed to understand neutrino oscillation physics?

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Thanks for the reply.Orodruin said:It is not a question of not knowing the masses of the flavour eigenstates, it is a matter of the flavour eigenstates not having definite masses.

Why do we never heard the masses of each mass eigenstates like nu_1= x,nu_2=y and so on,but just the differences of mass squared?

I'd it because of the lack information we have, and later by the datas improvemen's we can find them?

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See for example the oscillation equations on the Wikipedia page. They contain the masses only in the form ##\Delta m_{ij}^2## which means ##m_i^2 - m_j^2##.kimmm said:just the differences of mass squared?

Neutrinos are subatomic particles that have very little mass and no electric charge. They are important in understanding the mass of the universe because they are the second most abundant particles after photons, and their mass plays a crucial role in shaping the structure of the universe.

The oscillation phenomenon in neutrinos refers to the change in flavor or type of neutrinos as they travel through space. Neutrinos have three different types or flavors, and they can switch between these flavors as they move, which provides evidence for their non-zero mass.

Scientists measure the mass of neutrinos through a variety of methods, including studying the shape of the cosmic microwave background radiation, observing the decay of radioactive materials, and measuring the energy of neutrinos emitted from nuclear reactions. These methods provide indirect evidence for the mass of neutrinos.

The discovery of the mass of neutrinos can have significant implications for our understanding of the universe and its evolution. It can help us better understand the formation of galaxies, the distribution of matter in the universe, and the role of neutrinos in the early stages of the universe. It can also have applications in particle physics and potentially lead to new technologies.

Yes, there are several ongoing experiments and studies focused on exploring the oscillation phenomenon in neutrinos. Some of the most well-known include the Super-Kamiokande experiment in Japan, the IceCube experiment in Antarctica, and the Daya Bay Reactor Neutrino Experiment in China. These experiments aim to gather more data and evidence to further our understanding of neutrino oscillations and their mass.

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