Casimir energy between two parallel plates and projective representation of Poincar\'e group

TL;DR

This paper explores the relationship between the Casimir energy between parallel plates and the projective representations of the Poincaré group, revealing how vacuum energy relates to symmetry algebra central charges.

Contribution

It demonstrates that Casimir energy arises from projective representations of the Poincaré group, linking vacuum energy to algebraic central charges in quantum field theory.

Findings

Casimir energy is connected to central charges in Poincaré algebra.

Projective representations explain the consistency of vacuum energy with symmetry.

Analysis uses a massless scalar field as an example.

Abstract

The Casimir effect is a physical manifestation of zero point energy of quantum vacuum. In a relativistic quantum field theory, Poincar\'e symmetry of the theory seems, at first sight, to imply that non-zero vacuum energy is inconsistent with translational invariance of the vacuum. In the setting of two uniform boundary plates at rest, quantum fields outside the plates have (1+2)-dimensional Poincar\'e symmtry. Taking a massless scalar field as an example, we have examined the consistency between the Poincar\'e symmetry and the existence of the vacuum enegy. We note that, in quantum theory, symmetries are represented projectively in general and show that the Casimir energy is connected to central charges appearing in the algebra of generators in the projective representations.