Quantum energy inequalities

The stress-energy tensor plays a special role in quantum field theory, i.e., the mathematical description of the physics of elementary particles. This theory successfully predicts results of experiments, e.g., in particle accelerators like the Large Hadron Collider (LHC). Here two high-energy particle beams travel almost at the speed of light before colliding inside particle detectors. The detectors measure quantities like the particle’s speed, mass, and energy – from which physicists can determine a particle’s identity. Therefore, measurements of energy play an essential role in particle physics.

The energy density (energy per volume) also plays a crucial role in Einstein’s theory of General Relativity, which is fundamental to our present understanding of the geometric properties of space and time, or “spacetime”. It links the curvature of spacetime (and thus the gravitational force) to the energy density of the matter. The energy density therefore influences the structure of space and time. In particular, certain conditions on the energy density (“energy inequalities”) guarantee the absence of exotic spacetime geometries, like time machines, wormholes and warp drives.

Energy density in the Ising model.

Energy density in the Ising model.

These energy inequalities are known to be fulfilled in macroscopic physics. Unfortunately, they cannot hold in the world of elementary particle physics. Instead, one hopes that so called “quantum energy inequalities” (QEIs) hold, which would suggest that the above-mentioned results – such as absence of wormholes – can still hold for realistic matter.

Our goal is to clarify whether, and under which conditions, QEIs can hold in Quantum Field Theory. So far, they have been obtained only in very simple situations, excluding any interaction between the particles involved. By contrast, we investigate QEIs specifically in situations where interaction (scattering) between particles is present; we will study how they depend on the strength and specific type of interaction. To that end, we will first consider simplified models of particle physics, which capture only some aspects of the physical situation but are easier to handle mathematically. We will then progressively generalize the type of scattering described. The aim is to arrive at QEIs that are valid for a wide range of interaction types, general enough so that they are expected to hold for physically realistic models.