Quantized collision invariants on the sphere

We show that a measurable function g:Sᵈ⁻¹, with d≥ 3, satisfies the functional relation {equation*} g(ω)+g(ω_*)=g(ω')+g(ω_*'), {equation*} for all admissible ω,ω_*,ω',ω_*'ᵈ⁻¹ in the sense that {equation*} ω+ω_*=ω'+ω_*', {equation*} if and only if it can be written as {equation*} g(ω)=A+B·ω, {equation*} for some constants A∈ R and Bᵈ. 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.