The trace identity and the planar Casimir effect

摘要

The familiar trace identity associated with the scale transformation x~μ → x′~μ = e~(-λ)x~μ 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 φ(t, x~1, x~2 = -a) = 0 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 φ(t, x~1 = -a) = 0 holds and (ⅱ) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where φ(t, x~1, x~2 = -a) = 0, 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 et al [Ann. Phys. (N.Y.), 165, 162 (1985)] is used to incorporate the said boundary condition into the generating functional for the connected Green's functions.