we obtain the equations of motion for the density and spin fluctuations.

We find that four long wavelength ($|k|v_{T} \ll \Omega_{int}$), low frequency ($\omega \ll \Omega_{int}$), spin modes propagate in this system. These modes have a quadratic dispersion, characteristic of ferromagnetic systems, but have a $k=0$ offset. Two of the modes are transverse spin waves, and the other two are quadrupolar modes where the longitudinal magnetization fluctuates along with the transverse spin fluctuations ($\langle S_\mu S_\nu\rangle -\langle S_\nu S_\mu\rangle$).

The spin modes are given by the equation $1 = -(c_{0} + c_{2})\int\frac{d\vec{p}}{(2\pi)^{3}}\frac{f_{0} + \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0}}{\omega +\Omega_{int} + q - \frac{\vec{k}\cdotp\vec{p}}{m}}$ [@landau]. The transverse spin modes have dispersion $\omega(k) = \frac{k^{2}v^{2}_{T}}{2 \Omega_{int}} - q$. The fact that this has a negative energy at $k=0$ signifies that in the presence of a quadratic Zeeman shift, the initial state is not the thermodynamic ground state. One can decrease the energy by rotating the spins to be transverse to the magnetic field. However the frequency stays real, meaning that the initial state is metastable, and transverse spin will not be spontaneously generated.

Meanwhile, the quadrupolar modes have dispersion $\omega(k) = -4c_{2}n_{0} + \frac{k^{2}v^{2}_{T}}{2 \Omega_{int}}$. In a gas with positive $c_{2}$, the frequency vanishes at a 