Harmonic Analysis and Mean Field Theory

TL;DR

This work develops a unified, representation-theoretic framework to perform harmonic analysis for the Euclidean conformal group, enabling efficient computation of conformally invariant pairings, shadow transforms, and the Plancherel measure for arbitrary Lorentz structures. It presents two complementary computational strategies: a weight-shifting-operator approach yielding recursion relations and an explicit Fourier-space algorithm that diagonalizes the shadow transform, both applied to general MFT OPE data. The authors derive a general OPE-coefficient formula for Mean Field Theory with spinning operators, and provide detailed results in scalar, fermionic, current, and stress-tensor sectors in 4d and 3d, including comparisons with existing results and numerical bootstrap data. These methods fuse harmonic-analysis with practical computation of OPE data, offering tools for analytic bootstrap and large-N analyses in diverse spacetime dimensions and operator representations. The framework paves the way for systematic exploration of MFT data and conformal blocks across spins and dimensions, with potential applications to SYK-like models and holographic CFTs.

Abstract

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.