Positive Energy Conditions in 4D Conformal Field Theory

TL;DR

This paper proposes a spacetime-averaged weak energy inequality for 4D quantum field theories, deriving constraints on conformal field theories that are weaker than existing bounds and exploring implications for theory consistency.

Contribution

It introduces a new energy condition in 4D CFTs, providing weaker constraints than the conformal collider bounds and analyzing their implications using momentum-space Wightman functions.

Findings

The energy inequality constrains OPE coefficients in 4D CFTs.

The derived bounds are weaker than Hofman-Maldacena bounds.

In 3D CFTs, the constraint reduces to positivity of the energy-momentum tensor two-point function.

Abstract

We argue that all consistent 4D quantum field theories obey a spacetime-averaged weak energy inequality $\langle T^{00} \rangle \ge -C/L^4$, where $L$ is the size of the smearing region, and $C$ is a positive constant that depends on the theory. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 4D conformal field theories, and find that it places constraints on the OPE coefficients of the theory. The constraints we find are weaker than the "conformal collider" constraints of Hofman and Maldacena. We speculate that there may be theories that violate the Hofman-Maldacena bounds, but satisfy our bounds. In 3D CFTs, the only constraint we find is equivalent to the positivity of 2-point function of the energy-momentum tensor, which follows from unitarity. Our calculations are performed using momentum-space Wightman functions, which are remarkably simple functions of momenta, and may be of interest in their own right.