English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Motivated by a desire to understand quantum fluctuation energy densities andstress within a spatially varying dielectric medium, we examine the vacuumexpectation value for the stress tensor of a scalar field with arbitraryconformal parameter, in the background of a given potential that depends ononly one spatial coordinate. We regulate the expressions by incorporating atemporal-spatial cutoff in the (imaginary) time and transverse-spatialdirections. The divergences are captured by the zeroth- and second-order WKBapproximations. Then the stress tensor is "renormalized" by omitting the termsthat depend on the cutoff. The ambiguities that inevitably arise in thisprocedure are both duly noted and restricted by imposing certain physicalconditions; one result is that the renormalized stress tensor exhibits theexpected trace anomaly. The renormalized stress tensor exhibits no pressureanomaly, in that the principle of virtual work is satisfied for motions in atransverse direction. We then consider a potential that defines a wall, aone-dimensional potential that vanishes for $z<0$ and rises like $z^\alpha$,$\alpha>0$, for $z>0$. The full finite stress tensor is computed numericallyfor the two cases where explicit solutions to the differential equation areavailable, $\alpha=1$ and 2. The energy density exhibits an inverse lineardivergence as the boundary is approached from the inside for a linearpotential, and a logarithmic divergence for a quadratic potential. Finally, theinteraction between two such walls is computed, and it is shown that theattractive Casimir pressure between the two walls also satisfies the principleof virtual work (i.e., the pressure equals the negative derivative of theenergy with respect to the distance between the walls).

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