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Massive scalar field; trace identity; Casimir effect

The familiar trace identity associated with the scale transformationx^Μ→ x′^Μ = e^-λx^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, x¹, x² = -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, x¹ = -a) = 0 holds and (ii) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where Φ(t, x¹, x²= -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 Bordaget 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.

S G Kamath¹

1. Department of Mathematics, Indian Institute of Technology Madras, Chennai - 600 036, India

Published

February 2006

Manuscript received

27 May 2005

Manuscript revised

16 November 2005

Accepted

17 December 2005