New Methods for Conformal Correlation Functions

TL;DR

This paper introduces a unified method using embedding space formalism and a new uplift for quasi-primary operators to systematically compute all conformal correlation functions regardless of their Lorentz representations.

Contribution

It develops a novel uplift technique and tensorial functions that enable a comprehensive and recursive calculation of conformal correlation functions for all quasi-primary operators.

Findings

Unified treatment of all quasi-primary operators in conformal field theory.

Explicit computation of tensorial generalizations of Exton functions.

Systematic recursive method for calculating all correlation functions.

Abstract

The most general operator product expansion in conformal field theory is obtained using the embedding space formalism and a new uplift for general quasi-primary operators. The uplift introduced here, based on quasi-primary operators with spinor indices only and standard projection operators, allows a unified treatment of all quasi-primary operators irrespective of their Lorentz group irreducible representations. This unified treatment works at the level of the operator product expansion and hence applies to all correlation functions. A very useful differential operator appearing in the operator product expansion is established and its action on appropriate products of embedding space coordinates is explicitly computed. This computation leads to tensorial generalizations of the usual Exton function for all correlation functions. Several important identities and contiguous relations are also demonstrated for these new tensorial functions. From the operator product expansion all correlation functions for all quasi-primary operators, irrespective of their Lorentz group irreducible representations, can be computed recursively in a systematic way. The resulting answer can be expressed in terms of tensor structures that carry all the Lorentz group information and linear combinations of the new tensorial functions. Finally, a summary of the well-defined rules allowing the computation of all correlation functions constructively is presented.