Commutation relations for the electromagnetic field in the presence of dielectrics and conductors

Abstract

Motivated by recent investigations on the Casimir effect, we work out in detail the commutation relations satisfied by the quantized electromagnetic field in the presence of one or two dielectric slabs, with arbitrary dispersive and dissipative properties. In agreement with results derived by previous authors, we explicitly show that at all points in the empty region between the slabs, including their surfaces, the electromagnetic fields always satisfy free-field canonical equal-time commutation relations. This result is a consequence of general analyticity and detailed fall-off properties at large frequencies satisfied by the reflection coefficients of all real materials. It is also shown that this result is not obtained in the case of conductors, if the latter are modelled as perfect mirrors. In such a case, the free-field form of the commutation relations is recovered only at large distances from the mirrors. Failure of perfect-mirror boundary conditions to reproduce the correct form of the commutation relations near the surfaces of the conductors suggests that caution should be used when these idealized boundary conditions are used in investigations of proximity phenomena originating from the quantized electromagnetic field, like the Casimir effect.