Radiative Correction to The Casimir Energy with Mixed Boundary Condition in 2+1 Dimensions

TL;DR

This paper calculates the first-order radiative correction to the Casimir energy for a scalar field with mixed boundary conditions in 2+1 dimensions, using a systematic renormalization and regularization scheme, and compares it with previous results.

Contribution

It introduces a position-dependent counterterm and applies the Box Subtraction Scheme to compute finite radiative corrections for mixed boundary conditions in 2+1 dimensions.

Findings

The correction is finite and consistent with physical expectations.

The results differ from those with pure boundary conditions, highlighting boundary effects.

Comparison with previous boundary condition results shows agreement in special cases.

Abstract

In the present study, the zero- and first-order radiative correction to the Casimir energy for the massive and massless scalar field confined with mixed (Neumann-Dirichlet) boundary condition between two parallel lines in 2+1 dimensions for the self-interacting $\phi^4$ theory was computed. The main point in this study is the use of a special program to renormalize the bare parameters of the Lagrangian. The counterterm used in the renormalization program, which was obtained systematically position-dependent, is consistent with the boundary condition imposed on the quantum field. To regularize and remove infinities in the calculation process of the Casimir energy, the Box Subtraction Scheme as a regularization technique was used. In this scheme, two similar configurations are usually introduced, and the vacuum energies of these two configurations in proper limits are subtracted from each other. The final answer for the problem is finite and consistent with the expected physical basis. We also compared the new result of this paper to the previously reported results in the zero- and first-order radiative correction to the Casimir energy of scalar field in two spatial dimensions with Periodic, Dirichlet, and Neumann boundary conditions. Finally, all aspects of this comparison were discussed.