Negative Energy Densities in Quantum Field Theory With a Background Potential

TL;DR

This paper develops a method to calculate one-loop energy densities for scalar fields with background potentials, enabling tests of quantum energy conditions and revealing negative energy densities in specific scenarios.

Contribution

It introduces a general procedure for renormalized energy density calculations in quantum field theory with background potentials, including explicit examples and solvable models.

Findings

Negative energy densities found outside square barriers.

Divergent positive contributions inside barriers.

Exact solutions for sech^2 potentials in soliton studies.

Abstract

We present a general procedure for calculating one-loop ``Casimir'' energy densities for a scalar field coupled to a fixed potential in renormalized quantum field theory. We implement direct subtraction of counterterms computed precisely in dimensional regularization with a definite renormalization scheme. Our procedure allows us to test quantum field theory energy conditions in the presence of background potentials spherically symmetric in some dimensions and independent of others. We explicitly calculate the energy density for several examples. For a square barrier, we find that the energy is negative and divergent outside the barrier, but there is a compensating divergent positive contribution near the barrier on the inside. We also carry out calculations with exactly solvable $\sech^2$ potentials, which arise in the study of solitons and domain walls.