1.2.8. Flavor Waves

We have to specify the field before we can visualize the waves.

In the paper [Duan2009] the field is named as spin waves, which is in fact a field of rotating angles \(\phi(t,\mathbf x)\) of the flavor isospin around the direction of vacuum Hamiltonian in mass basis in this space. It is a flavor isospin field but is characterized by this rotating angle \(\phi(t,\mathbf x)\).

In general, we could think of the wave as a lot of internal spins, whose direction is defined by comparing them to the local Hamiltonian. The equation of motion is first order

\[(\partial_t + \hat v \cdot \boldsymbol \nabla) \mathbf s = \mathbf s \times \mathbf H,\]

which is a precession equation not a wave equation. We can not think of the spin itself as wave. For a constant \(\mathbf H\), Fourier transform of the equation shows that

\[\omega = \mathbf k \cdot \hat v,\]

and \(\tilde{\mathbf s}\) is parallel to \(\mathbf H\) in order to have solutions.

One of the interesting things to do is to connect this view of waves to the dispersion relation.

1.2.8.1. References and Notes

[Duan2009]Duan, H., Fuller, G. M., & Qian, Y.-Z. (2009). Neutrino Flavor Spin Waves. arXiv:0808.2046v1.

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