Conformal-helicity duality & the Hilbert space of free CFTs

Abstract

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.

6 pages + supplemental material

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