Quantized collision invariants on the sphere

We show that a measurable function $g:{\mathbb{S}}^{d-1}\to \mathbb{R}$, with $d\ge 3$, satisfies the functional relation \begin{equation*} g(ω)+g(ω_)=g(ω’)+g(ω_‘), \end{equation*} for all admissible $\omega ,{\omega }_{\ast },{\omega }^{\mathrm{\prime }},{\omega }_{\ast }^{\mathrm{\prime }}\in {\mathbb{S}}^{d-1}$ in the sense that \begin{equation*} ω+ω_=ω’+ω_’, \end{equation*} if and only if it can be written as \begin{equation*} g(ω)=A+B\cdotω, \end{equation*} for some constants $A\in \mathbb{R}$ and $B\in {\mathbb{R}}^d$. Such functions form a family of quantized collision invariants which play a fundamental role in the study of hydrodynamic regimes of the Boltzmann–Fermi–Dirac equation near Fermionic condensates, i.e., at low temperatures. In particular, they characterize the elastic collisional dynamics of Fermions near a statistical equilibrium where quantum effects are predominant.