# Schroediger Equation by Neutrino Oscillations

Hi all!

This question concerns flavour changing oscillations. Let's narrow it down to the neutrino case, where we have additionally the violation of lepton numbers. So electron and muon neutrinos naturally follow the relativistic Dirac equation:
$(p\!\!\!/ + m_e ) \nu_e = 0$ and $(p\!\!\!/ + m_{\mu} ) \nu_{\mu} = 0$ with $\nu = (\frac{1-\gamma_5}{2}) u_2$, where $u_2$ the standard spin down Dirac spinor for particles.
Now when we consider oscillations, we introduce the mass matrix $H_{mass} = \left( \begin{array}{cc} m_{e e} & m_{e \mu} \\ m_{\mu e} & m_{\mu \mu} \end{array} \right)$ and the flavour states apparently follow this strange form of Schroedinger equation:
$i\frac{\partial}{\partial t} \left( \begin{array}{c} \nu_e \\ \nu_{\mu} \end{array} \right) = H_{mass} \left( \begin{array}{c} \nu_e \\ \nu_{\mu} \end{array} \right)$
I've seen quoted that due to the same space-time dependence we can use the Schroedinger equation even for Dirac neutrinos. I'm clueless to what is going on here and would appreciate any help.

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First you need to correct those Dirac equation. Dirac eqn. does not involve ##m^2##, it involves only a single power of mass and those equations are two component eqn, so you should not use ##\not{p}## rather use ##\sigma.p##.

As to why you have a Schrodinger type eqn. is because for writing the time development you must write the flavor states in term of mass eigenstates. This time evolution will be governed by a unitary operator which will contain hamiltonian. It is similar to processes in quantum physics, where energy levels are mixed because of an interaction with some external field (magnetic field). So this mixing process has nothing to do with those Dirac equations.

Yes the mass squared term was an unfortunate copy&paste mistake, I've corrected it. But $\nu_e$ and $\nu_{\mu}$ are supposed to be Dirac four-spinors, representing the mass eigenstates, and $H_{mass}$ a 8x8 matrix. I want to treat neutrinos as regular fermions and thus stick to the Dirac representation.

So if we were to write a langrangian for each one of the two neutrinos, how would this mixing show up? An interaction term perhaps? I am not familiar with introducing ad-hoc unitary operators to describe such behaviour. At the end of the day, isn't the time evolution of those fields solely governed by their relativistic Dirac Hamiltonian? Or is this only true for the mass eigenstates and the flavour eigenstates follow a different evolution law?

Matterwave
Gold Member
Neutrino oscillations are beyond the standard model of particle interactions. The standard electro-weak theory would not have any neutrino oscillations, either they would be all massless, or the flavor states would have definite mass.

But we found that the flavor states are not states of definite mass. Remember that the Dirac equation is basically a statement of energy-mass equivalence. Just like the Klein-Gordon equation took E^2=p^2+m^2 directly and quantized the operators, Dirac used his special method to "square root" this equation and produce his first-order in time Dirac equation.

When we add in the fact that now flavor states (eigenstates of the weak interaction Hamiltonian) are no longer eigenstates of the mass-energy Hamiltonian, we get neutrino oscillations. Because neutrinos travel at v~c, they can be described by an energy-mass relation of the form:

$$E=\sqrt{p^2+m^2}\approx p+\frac{m^2}{2p}$$

Because of this approximation, we are allowed to write the mass matrix as a 2 by 2 (or 3x3 if you have 3 flavors) matrix H=diag(m1^2/2p, m2^2/2p). We usually assume the different mass states have the same momentum.

To answer your question, then, we've basically constructed our own energy-mass relation that the neutrinos must obey so to speak, in terms of their flavor degrees of freedom. It just superficially looks like the Schroedinger equation...but so does the Dirac equation:

$$i\hbar\frac{\partial}{\partial t}\psi(x,t)=H\psi(x,t)\quad H=\beta m+\vec{p}\cdot\vec{\alpha}$$

All of these laws are various forms of E^2=m^2+p^2 under different limits or different conditions. Schroedinger's is for m>>p, the neutrino's is for m<<p, and the Klein-Gordon equation and Dirac equation are valid for all p.

Yes the mass squared term was an unfortunate copy&paste mistake, I've corrected it. But $\nu_e$ and $\nu_{\mu}$ are supposed to be Dirac four-spinors, representing the mass eigenstates, and $H_{mass}$ a 8x8 matrix. I want to treat neutrinos as regular fermions and thus stick to the Dirac representation.
You do realize that the term ##1-\gamma_5## has it's lower two component zero, so you are supposed to write it as a two component eqn. when you act with it on ##u##, but if you want to write it as a four component one with lower two part zero, There is no problem with that either.
Anyway
So if we were to write a langrangian for each one of the two neutrinos, how would this mixing show up? An interaction term perhaps? I am not familiar with introducing ad-hoc unitary operators to describe such behaviour. At the end of the day, isn't the time evolution of those fields solely governed by their relativistic Dirac Hamiltonian? Or is this only true for the mass eigenstates and the flavour eigenstates follow a different evolution law?
The problem is that the eigenfunctions of the Hamiltonian are superpositions of neutrinos with different flavor numbers. We call them the mass eigenstates and they have the time development as
##|v(t)>=|v(0)>e^{-iEt}##
In a physical process, the neutrinos which are produced have definite flavor number. Their time development is more subtle because we must rewrite the flavor states in terms of mass eigenstates, whose time evolution is what I have written previously. The mis-match between the production of flavor states and the time evolution of mass states leads to an oscillation which is of course not related to the Dirac eqn. which describes the time evolution of individual particles.

@Matterwave

So let us do some math. We start with the approximation $E_i =\sqrt{p_i^2+m_i^2}\approx p_i+\frac{m_i^2}{2p_i} \approx E+\frac{m_i^2}{2E}$, whereby i denotes the mass eigenstate and E the total energy, indistinguishable for each state during a realistic experiment. Then we make use of the operator correspondence principle ($H_i = E_i \rightarrow i \frac{\partial}{\partial t}$) and write a schroedinger look-alike equation for each mass-state neutrino: $i\frac{\partial}{\partial t}\nu_i=H_i \nu_i$. Note that until this point $\nu_i$ is not a spinor! We then learn from experiment that there are two different mass eigenstates! So we may write a 2-component column vector of the mass neutrino: $\nu_m = \left( \begin{array}{c} \nu_1 \\ \nu_2 \end{array} \right)$ and the schroedinger look-alike equation: $i\frac{\partial}{\partial t}\nu_m=H_m \nu_m$. $H_m$ is a 2x2 diagonal matrix.
The next step is to rotate this vector with an orthogonal matrix $U$ and obtain the flavour eigenstates of the neutrino written again as a 2-component column vector: $\nu_f = \left( \begin{array}{c} \nu_e \\ \nu_{\mu} \end{array} \right) = U \nu_m$. Now $U$ happens to mix the flavour eigenstates, because the new hamiltonian is no longer diagonal: $H_f = U H_m U^{\dagger}$. From now on, proving that $\nu_f$ oscillates between $\nu_e$ and $\nu_{\mu}$ is logically straightforward.
I need a confirmation. Is the above reasoning correct? Am I missing something?
As a matter of fact, I understand that the flavour mixing was the first experimental result, but if I am to give a sound proof without having to introduce ad-hoc the rotation operator $U$, I would have to start with an approximate quantum hamiltonian for the mass eigenstates and observe that they are two of them, before I even talk about flavour mixing.

After we do all that, why are we allowed to return to the Dirac lagrangian and try to add new mass terms (!the interesting part!) for each flavour field? Wasn't the whole proof based on an approximation, which delivered scalar neutrino fields? Looking back at Pontecorvo's initial paper (Neutrino astronomy and lepton charge, 1962) my flavour-state hamiltonian in the above derivation $H_f$ is exactly his lagrangian extra mass term $L_{int}= \nu_{f}^C H_f \nu_{f}$ with $\nu_{f}^C$ the charge conjugated Dirac spinor. Are we permitted to commute between formulations so easily?

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You do realize that the term ##1-\gamma_5## has it's lower two component zero, so you are supposed to write it as a two component eqn. when you act with it on ##u##
That is not true in the Dirac representation.

The problem is that the eigenfunctions of the Hamiltonian are superpositions of neutrinos with different flavor numbers. We call them the mass eigenstates and they have the time development as
##|v(t)>=|v(0)>e^{-iEt}##
Why do they follow this particular time development? Unless you propose an approximation for some other relativistic wave equation, those mass eigenstates should definitely follow the Dirac time evolution.

The mis-match between the production of flavor states and the time evolution of mass states leads to an oscillation which is of course not related to the Dirac eqn. which describes the time evolution of individual particles.
But those individual particles were just put into collumns, so that new mass terms are introduced to explain the experimental results. For Pontecorvo e.g. this phenomenon is just another interaction term in the fermionic field lagrangian.

That is not true in the Dirac representation.
You are not allowed to use Dirac representation here, you have to work in weyl representation.
Why do they follow this particular time development? Unless you propose an approximation for some other relativistic wave equation, those mass eigenstates should definitely follow the Dirac time evolution.
It is precisely how the mass eigenstate evolves in time. There is no approximation regarding relativistic particle, the energy is simply √(p2+m2). Dirac equation has NOTHING to do with this mixing, See below
But those individual particles were just put into collumns, so that new mass terms are introduced to explain the experimental results. For Pontecorvo e.g. this phenomenon is just another interaction term in the fermionic field lagrangian.
It is not just like they are put into a column, the mass matrix mixes electron and muon neutrino. For example, at say t=0 you only have an electron neutrino and in course of time muonic component will develop which shows that flavor content of wave function changes with time. The general structure is determined by a unitary matrix (MNS matrix) which mixes flavor and mass states.
You can not use Dirac eqn. to describe these mixing process.

You are not allowed to use Dirac representation here, you have to work in weyl representation.
That is certainly something new. Weyl representation is connected to the Dirac by just a linear transformation. If by Weyl representation you mean 2-components vectors then you might be right, if the neutrino wavefunctions are scalars. But Weyl bispinors are something different.
It is precisely how the mass eigenstate evolves in time.
I insist, why is that so? Your time evolution is 100% a Schroedinger look-alike equation. Is this purely experimental for you?
Dirac equation has NOTHING to do with this mixing, See below
I think @Matterwave claimed that your time evolution is derived in a similar manner Dirac derived his but with an appropriate approximation. I'd appreciate, if you checked posts #4 and #5.
It is not just like they are put into a column, the mass matrix mixes electron and muon neutrino. For example, at say t=0 you only have an electron neutrino and in course of time muonic component will develop which shows that flavor content of wave function changes with time. The general structure is determined by a unitary matrix (MNS matrix) which mixes flavor and mass states.
That is all clear to me, PMNS mixes flavour and mass eigenstates. BUT the theory must have a langrangian. What is the kinematic term then?
Let's assume it is all seperate (=Neutrinos as fermions have their own dirac equations and the mixing is a seperate phenomenon). Where do the mathematics for the new theory come from? You clearly use some kind of Hamiltonian there. What is the connection with the old Dirac spinors?

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I actually don't have time right now to go through more details but this one has simple answer
I insist, why is that so? Your time evolution is 100% a Schroedinger look-alike equation. Is this purely experimental for you?
It's easy to see that a state with definite energy has e-iEt type dependence whether it is Dirac or Schrodinger state.
You may follow the book 'Massive neutrinos in physics and astrophysics' by Mohapatra and Pal for details regarding neutrino physics.

Matterwave
Gold Member
@Matterwave

So let us do some math. We start with the approximation $E_i =\sqrt{p_i^2+m_i^2}\approx p_i+\frac{m_i^2}{2p_i} \approx E+\frac{m_i^2}{2E}$, whereby i denotes the mass eigenstate and E the total energy, indistinguishable for each state during a realistic experiment. Then we make use of the operator correspondence principle ($H_i = E_i \rightarrow i \frac{\partial}{\partial t}$) and write a schroedinger look-alike equation for each mass-state neutrino: $i\frac{\partial}{\partial t}\nu_i=H_i \nu_i$. Note that until this point $\nu_i$ is not a spinor! We then learn from experiment that there are two different mass eigenstates! So we may write a 2-component column vector of the mass neutrino: $\nu_m = \left( \begin{array}{c} \nu_1 \\ \nu_2 \end{array} \right)$ and the schroedinger look-alike equation: $i\frac{\partial}{\partial t}\nu_m=H_m \nu_m$. $H_m$ is a 2x2 diagonal matrix.
The next step is to rotate this vector with an orthogonal matrix $U$ and obtain the flavour eigenstates of the neutrino written again as a 2-component column vector: $\nu_f = \left( \begin{array}{c} \nu_e \\ \nu_{\mu} \end{array} \right) = U \nu_m$. Now $U$ happens to mix the flavour eigenstates, because the new hamiltonian is no longer diagonal: $H_f = U H_m U^{\dagger}$. From now on, proving that $\nu_f$ oscillates between $\nu_e$ and $\nu_{\mu}$ is logically straightforward.
I need a confirmation. Is the above reasoning correct? Am I missing something?

This looks fine to me.

Ok, so it was easier and less rigorous than I expected.
The next step is to look for the origins of this neutrino mass and to introduce proper mass terms in the lagrangian of the theory. The most popular ideas are: i) a sterile right handed neutrino: $N_R$ (singlet under SU(2) ) ii) a Majorana neutrino: $\frac{1}{\sqrt{2}}(\nu+\nu^c)$ iii)combinations of these. A proper example would be then: $\mathcal{L}_{mass} = \left( \begin{array}{c} \overline{\nu}_L & \overline{N}^c_R \end{array} \right) \left( \begin{array}{cc} 0 & m_D \\ m_D & m_{MR} \end{array} \right) \left( \begin{array}{cc} \nu^c_L \\ N_R \end{array} \right) + h.c.$, with $m_D$ the Dirac mass and $m_{MR}$ the Majorana mass.
One last question, how do the different flavour families fit into that picture? Do the previously mentioned masses (i.e. $m_{ee}$, $m_{\mu\mu}$ and $m_{e \mu}$) arise from this theory or we must simply write: $\nu_L= \left( \begin{array}{c} \nu_{e L} \\ \nu_{\mu L} \end{array} \right)$, $N_R= \left( \begin{array}{c} N_{e R} \\ N_{\mu R} \end{array} \right)$ and replace $m_D$ and $m_{MR}$ with the respective 4x4 flavour mixing matrices? At the end of the day, do we have a theory for the flavour transitions or $m_{ee}$, $m_{\mu\mu}$ and $m_{e \mu}$ are just experimental facts we just have to plug in?

PS: $m_{MR}$ is a "bare" mass and if I am not mistaken we haven't postulated any mechanism to explain its existence (!if and only if it truely exists!).

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I am not sure what you are trying to show here. If you want to introduce a Majorana neutrino to explain oscillation then you should know that Majorana neutrino don't contribute to oscillation. They do give two phases in MNS matrix but they don't contribute to oscillation. They show their effects through neutrinoless double beta decay. The lagrangian you have written down is used rather as a tool to give an explanation of See-Saw mechanism.

And yes, at the end of the day, you have to go for experiments to know the mass differences rather than mass (only mass differences are determined here).

Well that lagrangian was just an example. At this point I simply asked, if we are "allowed" mathematically to deal with both flavours in the same expression for a general neutrino mass candidate. Let me try to formulate it again. For example, when we write terms like: $\overline{\nu}_L m_D N_R$ or $\overline{N}^c_R m_{MR} N_R$, can they always imply the summation ${\underset{i,j}{\sum}} \overline{\nu}_{L,i} m^{ij}_D N_{R,j}$ resp. ${\underset{i,j}{\sum}} \overline{N}^c_{R,i} m^{ij}_{MR} N_{R,j}$, with i and j running over the flavour eigenstates? I know for sure that the first fundamental mass term, the Higgs-mass, can be expanded in that way in the lagrangian. What about the rest of the possible terms?

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And yes, at the end of the day, you have to go for experiments to know the mass differences rather than mass (only mass differences are determined here).
I am sorry for the bad formulation, but the question didn't concern the fact that the values of the oscillation masses or better mass differences are experimental parameters. I'd rather like to know if we have a field theoretic explanation, which permits those mixing masses $m_{e \mu}$. Writing lagrangian terms like $\overline{\nu}_{L, \mu} m^{e\mu}_{D} N_{R, e}$ is alright, but I'm wondering if there is a deeper underlying mechanism here...

PS: By the way, basic textbooks do not address this question and I've yet to encounter an expert on theoretical neutrino physics.

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I'd rather like to know if we have a field theoretic explanation, which permits those mixing masses $m_{e \mu}$.
What do you mean by it.

What do you mean by it.
Well I may not know exactly what I mean by that. Maybe a version of some new SSB similar to the Higgs' mechanism. The whole theory is pure phenomenology so far and I seek some theoretical background. Physics beyond the SM must provide some mathematical structure for neutrino oscillations.
If you follow my reasoning though and you are familiar with the SM mechanism for the "allowance" of the dirac mass terms, then I think it is not absurd to ask ourselves, which is the underlying theory of these new proposed terms like $\overline{\nu}_{L, \mu} m^{e\mu}_{D} N_{R, e}$ or $\overline{N}^c_R m_{MR} N_R$, which are either ad-hoc and phenomenological or speculative. Don't they seem "bare" to you?

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Well, we needed a right handed neutrino to make a Dirac mass term and then we needed a Majorana mass term to explain the smallness of neutrino masses. If we even avoid RH neutrino, we can generate Majorana mass term by some dimension-5 operators which contains ordinary higgs doublet and a large mass scale.

There are some other models which are based on horizontal abelian charges in which there are fermions and flavons, some GUT models based on SO(10) times a flavor symmetry which do provide a somewhat richer structure but they are still not very successful.

In the end, just like CKM matrix, you have to go for experiments to determine the angles of MNS matrix and mass differences with the help of which you can determine mee and other ones.
This is the way our science is (you can not determine electron mass using pure QFT, you need experiment).( I am ending this discussion now)

I am going to answer for future reference some of the open questions of this thread:

Regarding post #12:
i) Both m_D$m_D$ and $m_{MR}$ can indeed be extended to 2x2 (or 3x3 for all flavors) matrices. It is noticeable however that we either diagonalize each term independently and study the oscillations or place them together in the mass matrix of the seesaw-mechanism and diagonalize before introducing the flavour mixing. For example, Pontecorvo uses only $m_{MR}$ in his original paper and discusses the neutrino-antineutrino oscillation.
Majorana neutrino don't contribute to oscillation.
ii) Oscillations are dictated by the mass terms in the Lagrangian. Whether the neutrino is actually a Dirac or a Majorana particle is irrelevant.

Regarding posts #15 & #17:
Fukugita & Yanagida's Physics of Neutrinos and Applications to Astrophysics is an excellent reference book to look up for theories, which try to explain both the smallness of the Dirac mass as well the arising of the Majorana mass. I quote three possibilities:

• Majoron Model (SSB of the lepton number symmetry similar to the Higgs Mechanism)
• Mass induced by radiactive Corrections (extra scalar field in the Higgs potential)
• Various BSM Models (Horizontal Symmetries, Peccei-Quinn symmetry, SO(10), Froggatt-Nielsen, to name a few)

The main question of this thread, namely the derivation of a Schroediger equation for the neutrino flavour mixing, is answered so unless somebody wants to add something, I consider it closed.