= Kac collision operator
{c}
{title2=$Q=\binom N2^{-1}\sum_{i<j}(2\pi)^{-1}\int U_{ij,\theta}\,d\theta$}
For $N\geq2$, $U_{ij,\theta}F=F\circ R_{ij,\theta}$ uses the genuine two-coordinate rotation $(v_i,v_j)\mapsto(v_i\cos\theta+v_j\sin\theta,-v_i\sin\theta+v_j\cos\theta)$. Average it uniformly over angles and pairs to obtain $Q$. Rotation invariance of <Lebesgue measure> makes $Q$ a bounded <self-adjoint operator> of <norm> one on $L^2(\mathbb R^N)$. Each pair average is an <orthogonal projection>, but their full average is not in general a projection.
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