Implications of Conformal Invariance for Quantum Field Theories in $d>2$

TL;DR

This paper explores the structure of conformally invariant quantum field theories in dimensions greater than two, analyzing correlation functions, energy-momentum tensor forms, and implications for conformal symmetry breaking.

Contribution

It generalizes the form of three-point functions and central charge concepts to higher dimensions and discusses positivity and renormalization group constraints.

Findings

Three linearly independent forms of the energy-momentum tensor three-point function in d>2.

Positivity conditions for energy density expectation values derived.

Proposed equations relating broken conformal invariance to correlation functions.

Abstract

Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions {\absit d} are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general {\absit d} compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to {\absit d} larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions.\hfill\break Talk presented at the XXVII Ahrenshoop International Symposium.