Energy Inequalities in Interacting Quantum Field Theories

Abstract

The classical energy conditions, originally motivated by the Penrose-Hawking singularity theorems of general relativity, are violated by quantum fields. A reminiscent notion of such conditions are the so called quantum energy inequalities (QEIs), which are however not known to hold generally in quantum field theory.Here we present first steps towards investigating QEIs in quantum field theories with self-interaction. IntroductionOne of the fundamental observables both in quantum and in classical field theory is the stress-energy tensor T αβ . It has a special role in general relativity, as the Einstein equations link the curvature of spacetime to the distribution of matter throughout it. Certain positivity conditions on the stress-energy tensor (e.g., the so called "weak energy condition"), which are fulfilled by many classical matter fields, imply severe constraints on exotic spacetime geometries. They also enter the Penrose and Hawking singularity theorems [19], positive mass theorems [22], and Hawking's chronology protection results [18], among many others. However, in quantum field theory (QFT) these energy conditions are violated; the energy density can have negative expectation values. This raises the question whether the energy conditions on the matter in the assumptions of the singularity theorems are compatible with quantum matter, or whether quantum fields allow the existence of "exotic" spacetimes like time machines, wormholes, warp drives. To exclude these scenarios, there must be constraints to the extent in which quantum fields can cause negative energy densities. These constraints are called "Quantum Energy Inequalities" (QEIs). First investigated by Ford [15], they are reminiscent of classical

An approximate local modular quantum energy inequality in general quantum field theory

“…Here, one uses the divergencefreeness of the expected stress-energy tensor, as mentioned in Section 3. Furthermore, one can use (A.6) and (A 16[34] to conclude that pW pu ' vqΩ, HW pu ' vqΩq " 1 2 pxu, Quy `xv, vyq , (A.72) a well-known result.…”

“…While there are numerous results on QWEIs and also averaged energy inequalities for linear quantum field theories, including such on general spacetimes, there is apparently little on locally averaged lower bounds on energy expectation values for general and interacting quantum field theories. The results in this direction so far have been relatively sparse [8,7,16],…”