The trace identity and the planar Casimir effect

TL;DR

This paper investigates how the trace identity related to scale transformations is violated by boundary conditions in a 2+1 dimensional scalar field, revealing an anomaly on a single Dirichlet plate.

Contribution

It demonstrates the violation of the trace identity in 2+1 dimensions with boundary conditions and introduces a method to incorporate boundary effects into Green's functions.

Findings

Trace identity violated on a Dirichlet plate in 2+1 dimensions

Trace identity respected in free space and 1+1 dimensions

Anomalous term coefficient equals the canonical scale dimension 1/2

Abstract

The familiar trace identity associated with the scale transformation xxxx on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition xxxx is imposed.It is however respected in : i. 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition xxxx holds; and, ii. in 2 + 1 dimensions in free space, i.e. the unconstrained configuration.On the plate where xxxx, the modified trace identity is shown to be anomalous with a numerical coefficient for the anomalous term equal to the canonical scale dimension viz.1/2. The technique of Bordag,Robaschik and Wieczorek [5] is used to incorporate the said boundary condition into the generating functional for the connected Green's functions. Note: The xxxx in the abstract above refer to symbols that are available in the abstract of the paper.