Stress tensor for a scalar field in a spatially varying background potential: Divergences, “renormalization”, anomalies, and Casimir forces

Abstract

Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporalspatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth-and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for z < 0 and rises like z α , α > 0, for z > 0. Previously, the stress tensor had been computed outside of the wall, whereas now we compute all components of the stress tensor in the interior of the wall. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, α = 1 and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

Vacuum energy density and pressure near a soft wall

“…The renormalization problem has been investigated separately [25,27] for a general potential; [27] uses higher-order WKB approximations, with special attention to the application to the power wall. Precise calculation of the renormalized stress tensor, including contributions from the non-WKB region of the spectrum, are in process.…”

“…Calculations so far (here and in [10,16,27]) deal primarily with small values of α and hence with the extraneous divergences near z = 0 caused by the singularity in the potential there. For very large values of α these divergences must disappear, and the stress tensor must resemble that of a hard wall near z = 1.…”

“…Detailed examination [12,14] showed that this problem existed for flat boundaries also, and that it was related to the direction dependence of vacuum energy as regularized by point-splitting [19]. That is the motivation for the study of the soft wall [10,[14][15][16]27].…”

“…Here we have not reviewed papers on scalar quantum field theory in general background scalar potentials and the resulting issues of renormalization, since that topic relates more to [27] and possible later papers. Nor have we listed the large body of papers on fields interacting with slabs of material (such as dielectrics) or delta-function potentials; such models are not truly "soft" by our definition, since boundary divergences remain.…”

“…We now have 27) which is always positive. On the other hand, the calculation of the expectation values can be based from the outset on the "cylinder kernel", a certain Green function for the equation…”