Vacuum energy density and pressure of a massive scalar field

TL;DR

This paper calculates the stress tensor components of a massive scalar field in 1+1 dimensions, clarifying boundary effects and exploring regularization techniques like Pauli-Villars for Casimir energy computations.

Contribution

It generalizes previous work by Hays and Fulling, providing exact solutions and a detailed analysis of boundary contributions and regularization methods for the Casimir effect.

Findings

Explicit Green function construction using classical paths

Distinction of boundary and confinement contributions to energy density

Discussion of Pauli-Villars regularization for finiteness

Abstract

With a view toward application of the Pauli-Villars regularization method to the Casimir energy of boundaries, we calculate the expectation values of the components of the stress tensor of a confined massive field in 1+1 space-time dimensions. Previous papers by Hays and Fulling are bridged and generalized. The Green function for the time-independent Schrodinger equation is constructed from the Green function for the whole line by the method of images; equivalently, the one-dimensional system is solved exactly in terms of closed classical paths and periodic orbits. Terms in the energy density and in the eigenvalue density attributable to the two boundaries individually and those attributable to the confinement of the field to a finite interval are distinguished so that their physical origins are clear. Then the pressure is found similarly from the cylinder kernel, the Green function associated most directly with an exponential frequency cutoff of the Fourier mode expansion. Finally, we discuss how the theory could be rendered finite by the Pauli-Villars method.