Harmonic Analysis and Mean Field Theory

TL;DR

This paper reviews harmonic analysis for the Euclidean conformal group, introduces efficient computational methods, and derives a general formula for OPE coefficients in Mean Field Theory applicable to various operators and dimensions.

Contribution

It provides new computational techniques and a general formula for OPE coefficients in Mean Field Theory for spinning operators in different dimensions.

Findings

Derived a general formula for OPE coefficients in MFT for arbitrary spinning operators.

Introduced two efficient methods for harmonic analysis computations: weight-shifting operators and Fourier space approach.

Applied the formula to examples including fermions, seed operators, currents, and stress-tensors in 3d and 4d.

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.