Conformal Covariance and Positivity of Energy in Charged Sectors

TL;DR

This paper demonstrates that in Chiral Conformal Quantum Field Theories, the positivity of energy extends from the vacuum to charged sectors due to diffeomorphism covariance, emphasizing the local nature of energy positivity.

Contribution

It shows that energy positivity in Chiral Conformal QFTs is a local property and extends to charged sectors under diffeomorphism covariance and energy positivity in the vacuum.

Findings

Energy density remains bounded from below when smeared with nonnegative test functions.

Diffeomorphism covariance ensures positivity of energy in all sectors.

The proof uses recent methods for smearing stress-energy tensors on nonsmooth functions.

Abstract

It has been recently noted that the diffeomorphism covariance of a Chiral Conformal QFT in the vacuum sector automatically ensures M\"obius covariance in all charged sectors. In this article it is shown that the diffeomorphism covariance and the positivity of the energy in the vacuum sector even ensure the positivity of the energy in the charged sectors. The main observation of this paper is that the positivity of the energy -- at least in case of a Chiral Conformal QFT -- is a local concept: it is related to the fact that the energy density, when smeared with some local nonnegative test functions, remains bounded from below (with the bound depending on the test function). The presented proof relies in an essential way on recently developed methods concerning the smearing of the stress-energy tensor on nonsmooth functions.