Modified Weak Energy Condition for the Energy Momentum Tensor in Quantum Field Theory

TL;DR

This paper proposes a modified weak energy condition in quantum field theory, restricting the class of states considered to ensure positivity and consistency with free field theory results, especially in curved spacetime.

Contribution

It introduces a natural restriction on quantum states for the energy momentum tensor, leading to a modified weak energy condition that aligns with known free field theory outcomes.

Findings

Positivity conditions impose non-trivial inequalities on three-point function parameters.

The modified condition is satisfied in free theories, except near two dimensions.

It implies the positivity of the topological term coefficient in curved space.

Abstract

The weak energy condition is known to fail in general when applied to expectation values of the the energy momentum tensor in flat space quantum field theory. It is shown how the usual counter arguments against its validity are no longer applicable if the states $|\psi \r$ for which the expectation value is considered are restricted to a suitably defined subspace. A possible natural restriction on $|\psi \r$ is suggested and illustrated by two quantum mechanical examples based on a simple perturbed harmonic oscillator Hamiltonian. The proposed alternative quantum weak energy condition is applied to states formed by the action of scalar, vector and the energy momentum tensor operators on the vacuum. We assume conformal invariance in order to determine almost uniquely three-point functions involving the energy momentum tensor in terms of a few parameters. The positivity conditions lead to non trivial inequalities for these parameters. They are satisfied in free field theories, except in one case for dimensions close to two. Further restrictions on $|\psi \r$ are suggested which remove this problem. The inequalities which follow from considering the state formed by applying the energy momentum tensor to the vacuum are shown to imply that the coefficient of the topological term in the expectation value of the trace of the energy momentum tensor in an arbitrary curved space background is positive, in accord with calculations in free field theories.