One-Loop Correction to The Casimir Energy in Lifshitz-Like Theory

TL;DR

This paper calculates the one-loop radiative correction to the Casimir energy for Lifshitz-like scalar fields under various boundary conditions, using a systematic renormalization approach that accounts for boundary effects in 3+1 dimensions.

Contribution

It introduces a position-dependent renormalization scheme for Casimir energy corrections in Lifshitz-like theories with multiple boundary conditions.

Findings

The corrections are consistent with physical expectations.

Casimir energies vary significantly with boundary conditions.

The sign and magnitude of the energies depend on boundary types.

Abstract

In the present article, Radiative Correction (RC) to the Casimir energy was computed for the self-interacting massive/massless Lifshitz-like scalar field, confined between a pair of plates with Dirichlet and Mixed boundary conditions in $3+1$ dimensions. Moreover, using the results obtained for the Dirichlet Casimir energy, the RC to the Casimir energy for Periodic and Neumann boundary conditions were also draw outed. To renormalize the bare parameters of the Lagrangian, a systematic perturbation expansion was used in which the counterterms were automatically obtained in a position-dependent manner. In our view, the position dependency of the counterterm was allowed, since it reflected the effects of the boundary condition imposed or the background space in the problem. All the answers obtained for the Casimir energy were consistent with well-known physical expects. In the language of graphs, the Casimir energy for the massive Lifshitz-like scalar field confined with four boundary conditions (Dirichlet, Neumann, Mixed, and Periodic) was also compared to each other, and as a concluding remark, the sign and magnitude of their values were discussed.