A Quantum Focussing Conjecture

TL;DR

This paper introduces a universal quantum focusing conjecture linking entropy and geometry, extending classical bounds to quantum regimes and implying a new quantum null energy condition with potential implications for quantum gravity.

Contribution

It proposes a new quantum focusing conjecture unifying classical and quantum bounds, and derives a quantum null energy condition from it.

Findings

Formulation of a quantum focusing conjecture that generalizes classical theorems.

Derivation of a quantum Bousso bound from the conjecture.

Establishment of a quantum null energy condition in quantum field theory.

Abstract

We propose a universal inequality that unifies the Bousso bound with the classical focussing theorem. Given a surface $\sigma$ that need not lie on a horizon, we define a finite generalized entropy $S_\text{gen}$ as the area of $\sigma$ in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence $N$ orthogonal to $\sigma$, the rate of change of $S_\text{gen}$ per unit area defines a quantum expansion. We conjecture that the quantum expansion cannot increase along $N$. This extends the notion of universal focussing to cases where quantum matter may violate the null energy condition. Integrating the conjecture yields a precise version of the Strominger-Thompson Quantum Bousso Bound. Applied to locally parallel light-rays, the conjecture implies a Quantum Null Energy Condition: a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of this novel relation in quantum field theory.