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 :math:`\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 :math:`\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 .. math:: (\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 :math:`\mathbf H`, Fourier transform of the equation shows that .. math:: \omega = \mathbf k \cdot \hat v, and :math:`\tilde{\mathbf s}` is parallel to :math:`\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.