  We identify a means to explicitly construct primary operators of free
conformal field theories (CFTs) in spacetime dimensions $d=2,~3$, and $4$.
Working in momentum space with spinors, we find that the
$N$-distinguishable-particle Hilbert space $\mathcal{H}_N$ exhibits a $U(N)$
action in $d=4$ ($O(N)$ in $d=2,3$) which dually describes the decomposition of
$\mathcal{H}_N$ into irreducible representations of the conformal group. This
$U(N)$ is a natural $N$-particle generalization of the single-particle $U(1)$
little group. The spectrum of primary operators is identified with the
harmonics of $N$-particle phase space which, specifically, is shown to be the
Stiefel manifold $V_2(\mathbb{C}^N) = U(N)/U(N-2)$ (respectively,
$V_2(\mathbb{R}^N)$, $V_1(\mathbb{R}^N)$ in $d=3,2$). Lorentz scalar primaries
are harmonics on the Grassmannian $G_2(\mathbb{C}^N) \subset
V_2(\mathbb{C}^N)$. We provide a recipe to construct these harmonic polynomials
using standard $U(N)$ ($O(N)$) representation theory. We touch upon
applications to effective field theory and numerical methods in quantum field
theory.
