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Casimir energy between two parallel plates and projective representation of Poincaré group

The Casimir effect is a physical manifestation of zero point energy of quantum vacuum. In a relativistic quantum field theory, Poincaré 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 Poincaré symmtry. Taking a massless scalar field as an example, we have examined the consistency between the Poincaré 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.