Correlation Functions of the Energy Momentum Tensor on Spaces of Constant Curvature

TL;DR

This paper investigates the correlation functions of the energy momentum tensor on constant curvature spaces, deriving explicit expressions and analyzing their relation to conformal anomalies and the $c$-theorem in various dimensions.

Contribution

It provides a geometric, coordinate-independent formalism for correlation functions on constant curvature spaces and explores their connection to conformal anomalies and spectral representations.

Findings

Explicit expressions for correlation functions in free scalar, spinor, and vector theories.

Analysis of the role of conformal symmetries and Ward identities.

Demonstration that two-point functions are not directly related to the $a$ coefficient at non-coincident points.

Abstract

An analysis of one and two point functions of the energy momentum tensor on homogeneous spaces of constant curvature is undertaken. The possibility of proving a $c$-theorem in this framework is discussed, in particular in relation to the coefficients $c,a$, which appear in the energy momentum tensor trace on general curved backgrounds in four dimensions. Ward identities relating the correlation functions are derived and explicit expressions are obtained for free scalar, spinor field theories in general dimensions and also free vector fields in dimension four. A natural geometric formalism which is independent of any choice of coordinates is used and the role of conformal symmetries on such constant curvature spaces is analysed. The results are shown to be constrained by the operator product expansion. For negative curvature the spectral representation, involving unitary positive energy representations of $O(d-1,2)$, for two point functions of vector currents is derived in detail and extended to the energy momentum tensor by analogy. It is demonstrated that, at non coincident points, the two point functions are not related to $a$ in any direct fashion and there is no straightforward demonstration obtainable in this framework of irreversibility under renormalisation group flow of any function of the couplings for four dimensional field theories which reduces to $a$ at fixed points.