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So (7.55) can be rewritten as (7.58) This is in agreement with what Glauber and Lewenstein obtained by directly quantizing the macroscopic Maxwell equations for the case of a nondispersive and nonabsorptive dielectric medium [237]. So there is no extra noise in the displacement field coming from the atoms of the medium in this case. Rosewarne, however, claimed that there is such noise in the electric field [528]. He calculated the variance of the electric field in a scalar version of the Hopfield model and found that even in the case of weak coupling (G' ,W wo), there is a noise contribution from the atoms on the order of G,2 jwow. So it is even larger than the 4G,2 jw~ contribution of the medium to the dielectric permittivity. This extra noise is not accounted for when the medium is described only in terms of a dielectric permittivity. Then how can we use the idea of dielectric permittivity in quantum electrodynamics without being inconsistent Let us examine the electric field in more detail. In terms of the displacement field, the electric field is given by

(7.59)

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As D has no noise contribution from the medium up to second order, the contribution Rosewame found can come only from the polarization P. But the polarization (7.2) vanishes whenever one is not at one of the positions occupied by an atom of the medium. So except for these positions, there should be no noise contribution from the medium just as there is not for D. Rosewame's E had no such discontinuities, because he assumes a continuous distribution of atoms. This means that the dynamical variables he uses are supposed to be macroscopic averages from the beginning. How can we define a macroscopic average Lorentz introduced macroscopic averages to show how Maxwell's equations in material media can be obtained from more fundamental equations for the electromagnetic fields in the vacuum and the constituent charged particles that make up the media. This average was over "physically infinitesimal" volume elements, which were yet large enough to contain many atoms. There are some problems with Lorentz's definition of macroscopic average. Perhaps the most important for us is that in many physical situations there is no physically infinitesimal volume that is both macroscopically small and yet contains a large number of atoms. In optics, for example, the volume element cannot exceed 109 A3~ otherwise, optical oscillations would be averaged to zero. But the addition of a single extra electron to such a volume would cause an enormous change in the charge density: about 160 C/m 3 -by no means an infinitesimal change! Many alternative definitions of macroscopic averaging have been proposed in the literature.6 A well-established trend is to regard macroscopic averaging as just a

sort of ensemble averaging. Robinson [524, 525] points out, however, that the aim of doing macroscopic averages is to discard unwanted microscopic detail. This is different from the ensemble averages of statistical physics, for there we do an average only because our knowledge of the macroscopic state of the system is incomplete. Robinson showed that the best way to get rid of the unwanted microscopic detail from a given dynamical variable is to eliminate high-frequency components from its spatial Fourier expansion. We will adopt his idea here, as it seems to have "hit the nail on the head" as far as macroscopic averaging is concerned. As the cavity has a finite length, we can expand the delta functions in (7.2) into a Fourier series. Moreover, as the atoms of the medium can be everywhere inside the cavity except at the positions of the cavity mirrors, this Fourier series will be a sine series; that is,

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