A Lower Bound on the Energy Density in Classical and Quantum Field Theories

TL;DR

This work develops a general framework connecting global energy positivity to local energy bounds across classical, quantum, and gravitational field theories. It uses an information-theoretic construction, M, to convert global stability into a pointwise bound, yielding a local energy condition in 1D classical theories, a covariant improved dominant energy bound, and a semilocal quantum bound tied to entanglement entropy, including the QNEC and conjectured QDEC. The approach extends to curved spacetimes and higher dimensions, where surface data and nonlocal entropic terms govern the bounds, suggesting a deep link between energy conditions and information theory. The results offer a unified perspective on energy stability that bridges classical fields, QFT, and gravity, with potential implications for singularity theorems and causality constraints.

Abstract

A novel method for deriving energy conditions in stable field theories is described. In a local classical theory with one spatial dimension, a local energy condition always exists. For a relativistic field theory, one obtains the dominant energy condition. In a quantum field theory, there instead exists a quantum energy condition, i.e. a lower bound on the energy density that depends on information-theoretic quantities. Some extensions to higher dimensions are briefly discussed.