Solve Neutrino Oscillations Homework: Eigenspinors & Particle Masses

In summary, the conversation discusses the creation of a homework problem involving neutrino oscillations and the field equations for two spinor fields. It is shown that the eigenspinors of the hermitian matrix representing the field equations correspond to particles with mass m_1 and m_2. However, the spinor fields do not satisfy the usual Dirac equation and the key to interpreting them as particles lies in the derivative term of the equation of motion. The derivative term acts as the identity on the eigenspinors of the mass matrix.
  • #1
Pietjuh
76
0
I have to make a homework problem about neutrino oscillations, but I already don't know how to answer the first question.
Let [tex]\Psi_i[/tex], i = 1,2 be two spinor fields, with field equation
[tex]\gamma^{\mu}\partial_{\mu}\Psi_i = - \sum_{j=1}^2 M_{ij} \Psi_j[/tex]

where [tex]M_{ij}[/tex] is a hermitian matrix. Suppose this matrix has eigenvalues m1 and m2. Show that the eigenspinors of this matrix represent particles with mass [tex]m_1 = \frac{\hbar\mu_1}{c}[/tex] and [tex]m_2 = \frac{\hbar\mu_2}{c}[/tex]

I know that each of the two spinorfields have to satisfy the dirac equation. But the field equations of these spinorfields are coupled equations, so I can't just make the correspondence. From the dirac equation I know that:

[tex]i \gamma^{\mu}\partial_{\mu}\Psi_i - \frac{m_i c}{\hbar}\Psi_i = 0[/tex]

This gives me that the field equations equal to:

[tex]
\sum_{j=1}^2 M_{ij}\Psi_j = - \frac{m_i c}{i\hbar}\Psi_i
[/tex]

But if we now compose a new vector, consisting of the two spinorfields joined in one, we get the following:

[tex]
M \Psi = - \frac{c}{i\hbar} \left(\begin{array}{cc} m_1 e_4 & 0 \\
0 & m_2 e_4\end{array}\right)
[/tex]

But I don't really know how we can determine from this the masses of the particles. Can someone give me a hint?
 
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  • #2
The spinor fields [tex] \Psi_i [/tex] do not satisfy the usual Dirac equation. You have their equations of motion given to you. However, in order to interpret things as particles, you would like to put the equations of motion into the Dirac form. The key observation is that the derivative term in the equation of motion is just the identity in the 1,2 space. Hint: how does the derivative term act on the eigenspinors of the mass matrix?

Let me know if you need further help.
 

What are neutrino oscillations?

Neutrino oscillations refer to the phenomenon where neutrinos, which are subatomic particles, change between different "flavors" as they travel through space. These flavors include electron, muon, and tau neutrinos. This behavior is possible due to the quantum mechanical nature of neutrinos.

What are eigenspinors?

Eigenspinors are mathematical objects used to describe the quantum states of particles. In the context of neutrino oscillations, eigenspinors are used to represent the different flavors of neutrinos and how they change over time.

How do eigenspinors relate to particle masses?

In the context of neutrino oscillations, eigenspinors are used to describe the different mass states of neutrinos. Each mass state corresponds to a different eigenvalue of the eigenspinor matrix.

What is the significance of neutrino oscillations?

Neutrino oscillations have important implications for our understanding of the Standard Model of particle physics. They provide evidence for the fact that neutrinos have mass, which was previously thought to be massless. Neutrino oscillations also have potential applications in fields such as astrophysics, where they may help us better understand the behavior of neutrinos in extreme environments.

How are neutrino oscillations studied and measured?

Neutrino oscillations are studied and measured through experiments such as the Super-Kamiokande and IceCube detectors, which are able to detect the different flavors of neutrinos and measure their oscillation patterns. Other experiments such as the MINOS and T2K collaborations also study neutrino oscillations using particle accelerators.

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