The Quantum Focusing Conjecture and the Improved Energy Condition

By rearranging its terms, the Quantum Focusing Conjecture (QFC) can be viewed as a quantum energy condition, and we can consider various limits. A recent restricted version is a limiting form where the quantum focusing vanishes $\mathrm{\Theta }\to 0$, and has been proven for Braneworld scenario. As a result, we derive an improved quantum null energy condition (INEC) $T_{kk}\ge \frac{\mathrm{\hslash }}{2\pi \mathcal{A}}(S_{out}^{\mathrm{\prime }\mathrm{\prime }}-\frac{1}{2}\theta S_{out}^{\mathrm{\prime }})$, that can be proven with field theory techniques. We sketch the beginning of a proof, and briefly discuss possible interpretations in the absence of one.