Conformally Invariant Green Functions of Current and Energy-Momentum Tensor in Spaces of Even Dimension D >= 4

TL;DR

This paper investigates conformally invariant Green functions for current and energy-momentum tensors in even-dimensional spaces, deriving new propagator expressions that satisfy Ward identities and are suitable for conformal QED and gravity.

Contribution

It introduces new conformally invariant propagator formulas with integrable singularities for current and energy-momentum tensors in even dimensions, advancing the understanding of conformal field theories.

Findings

Derived new expressions for < j_ield> and < T_{ield}> propagators.

Proved these propagators satisfy Ward identities.

Showed integrals over internal lines are conformally invariant and convergent.

Abstract

We study the conformally invariant quantum field theory in spaces of even dimension D >= 4. The conformal transformations of current j_\mu and energy-momentum tensor T_{\mu\nu} are examined. It is shown that the set of conformal transformations of particular kind corresponds to the canonical (unlike anomalous) dimensions l_j=D-1 and l_T=D of those fields. These transformations cannot be derived by a smooth transiton from anomalous dimensions. The structure of representations of the conformal group, which correspond to these canonical dimensions, is analyzed, and new expressions for the propagators < j_\mu j_\nu > and < T_{\mu\nu} T_{\rho\sigma}> are derived. The latter expressions have integrable singularities. It is shown that both propagators satisfy non-trivial Ward identities. The higher Green functions of the fields j_\mu and T_{\mu\nu} are considered. The conformal QED and linear conformal gravity are discussed. We obtain the expressions for invariant propagators of electromagnetic and gravitational fields. The integrations over internal photon and graviton lines are performed. The integrals are shown to be conformally invariant and convergent, provided that the new expressions for the propagators are used.