# 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.

| Created with Sphinx and . | | Index |