Problem with neutrino oscillations

[ShayanJ]

The phenomena of neutrino oscillations (as I understand it) is based on the idea that neutrino mass eigenstates are not the same as the flavor eigenstates and being in a definite mass eigenstate means the paticle has no definite flavor and vice versa. But this doesn't make sense to me, because the quantum state of a particle is the combination of its state in several Hilbert spaces, the Hilbert space of complex functions over spatial coordinates, the flavor Hilbert space, etc. As long as the interactions in the Hamiltonian don't couple flavor with spatial coordinates, these Hilbert spaces are completely independent and have nothing to do with each other. So I don't understand this effect that arises from the supposed inter-dependency of these two independent Hilbert spaces. Can anyone make things clear?
Thanks

 

[Orodruin]

It is unclear to me which part is unclear to you. Are you saying that you do not understand how the mass eigenstates can be different from the flavour eigenstates (this is also the case in the quark sector) or are you saying that you do not understand how it can give rise to neutrino oscillations?

 

[ShayanJ]

I don't understand how this can give rise to neutrino oscillations. Because the only ways I know that different parts of a quantum state(I mean parts in different Hilbert spaces) can affect each other, is either with the mixing of these parts through a Hamiltonian that has a cross term between these Hilbert spaces or something like having a state that has to be either completely symmetric or completely anti-symmetric w.r.t. particle exchange that is the product of a spin state and a spatial state. Then the symmetry properties of one of these parts determines the symmetry properties of the other. But it doesn't seem to me that neutrino oscillations is of any of the above types.

 

[ShayanJ]

Also, of course those eigenstates are different! They're two sets of vectors in two different Hilbert spaces. It doesn't even make sense to ask the question whether they're different or not, let alone concluding a specific physical phenomenon on the basis of them being different!

 

[Orodruin]

The typical way of deriving the neutrino oscillation formula does not bother at all with the spatial part of the wave function. It only deals with flavour space. It is completely analogous to spin precession where you are interested in the z-component of the spin of a spin-1/2 particle in an external magnetic field in the x-direction. The interaction basis in that case is the z-basis and the evolution basis is the x-basis.

 

[Vanadium 50]

It's better viewed as interference than oscillations. To avoid retyping, let me point you here: https://www.physicsforums.com/threa...ery-low-energy-neutrinos.900104/#post-5664961

 

[Orodruin]

@ShayanJ I think you are deceiving yourself in some way from the $L$ dependence that the spatial wave function has anything to do with the oscillations (you can study the effects of spatial wave packets rather than plane waves, but in essence that just leads to decoherence and it is not really relevant to what you are looking at). The $L$ in the standard oscillation formula is a proxy for time as $L \simeq t$ (in units of $c = 1$) for ultrarelativistic neutrinos and does not a priori have anything to do with the spatial wave function. The mass eigenstates and flavour eigenstates span the same Hilbert space* and are related by a unitary rotation. The mass eigenstates are the propagation eigenstates that acquire a definite phase from the time evolution and the flavour eigenstates are the ones that produce a particular charged lepton via charged current interactions.

* The flavour states are not really well defined as they are not asymptotic states of the propagation Hamiltonian. It is a good approximation for ultrarelativistic neutrinos though.