Vacuum energy density and pressure near a soft wall

TL;DR

This paper develops a formalism for quantizing scalar fields near a soft wall potential, showing that the pressure anomaly present in idealized models does not occur outside the wall, and sets the stage for further regularization studies.

Contribution

It introduces a realistic soft wall model for scalar fields, providing a framework that avoids the pressure anomaly and facilitates numerical analysis of energy density and pressure.

Findings

Pressure anomaly does not occur outside the soft wall

Numerical calculations show consistent energy density and pressure outside the wall

Formalism established for quantization and approximation in soft wall backgrounds

Abstract

Perfectly conducting boundaries, and their Dirichlet counterparts for quantum scalar fields, predict nonintegrable energy densities. A more realistic model with a finite ultraviolet cutoff yields two inconsistent values for the force on a curved or edged boundary (the "pressure anomaly"). A still more realistic, but still easily calculable, model replaces the hard wall by a power-law potential; because it involves no a posteriori modification of the formulas calculated from the theory, this model should be anomaly-free. Here we first set up the formalism and notation for the quantization of a scalar field in the background of a planar soft wall, and we approximate the reduced Green function in perturbative and WKB limits (the latter being appropriate when either the mode frequency or the depth into the wall is sufficiently large). Then we display numerical calculations of energy density and pressure for the region outside the wall, which show that the pressure anomaly does not occur there. Calculations inside the wall are postponed to later papers, which must tackle regularization and renormalization of divergences induced by the potential in the bulk region.