Quantized collision invariants on the sphere

We show that a measurable function g: S^d−1 → ℝ, with d ≥ 3, satisfies the functional relation g(ω) + g(ω_∗) = g(ω^′) + g(ω^′∗), for all admissible ω, ω_∗, ω^′, ω^′∗ ∈ S^d−1 in the sense that if and only if it can be written as ω + ω_∗ = ω^′ + ω^′∗, g(ω) = A + B · ω, for some constants A ∈ ℝ and B ∈ 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.