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. References and Notes

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

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