9.4 INCOMING VACUUM MODES AND COMPLETENESS

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As they obey Sommerfeld's radiation condition, natural modes on their own do not form a complete set. So we cannot expand the quantized electromagnetic radiation field in terms of them alone. In this section we formulate this problem more precisely and suggest how they can be supplemented by a different sort of mode in a way that will make the total set complete. But before starting this discussion, it is convenient to introduce first a fascinating result that is not widely taught in electrodynamics lecture courses today: Although we normally use one scalar potential and a vector potential to describe the electromagnetic field, the same job can be done using only two real scalar potentials or, equivalently, a single complex scalar potential.

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13He was later to become the discoverer of the first elementary particle, the electron. 14See page 95, "Researches, 1880-84" in m of [596].

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Whittaker's scalar potentials: From a vector and a scalar potential to just two scalar potentials

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We know from elementary electrodynamics that a description in terms of electromagnetic potentials introduces redundant degrees of freedom, whose main consequence is giving us the possibility of choosing the gauge that suits us best. Can we use this redundancy to reduce the number of electromagnetic potentials needed to describe the fields Yes, we can. We will now show that this redundancy can be exploited to reduce the usual description in terms of a scalar potential and a vector (with three scalar components) potential to one involving only two scalar potentials. This is not only useful to simplify calculations, but also seems to have some fundamental significance, as we know that light has only two independent polarizations (see s 2 and 3). One can derive this representation in a general way, in the presence of matter [466] and without making specific assumptions about the cavity shape and so on. As we just wish to develop the basic ideas of such representation, however, we will assume vacuum and take a specific cavity geometry that will lead to a simple presentation of these ideas. Consider a cylindrical cavity geometry, where the surfaces of the cavity walls are everywhere parallel to the z-axis. Assuming that the cavity is just sitting in the vacuum, the frequency w = ck components of the fields satisfy the following Maxwell equations:

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\1 i\ B + ik E = 0, \1. E = 0, \1 i\ E - ik B = 0,

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(9.64)

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(9.65) (9.66) (9.67)

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\1 B =0,

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with the boundary conditions (see Appendix B)

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it i\ Els

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and it Bls = 0,

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(9.68)

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where it is the unit normal to a cavity wall and Is denotes that these quantities are to be evaluated on the cavity walls. To satisfy the vanishing divergence equations (9.65) and (9.67) automatically, we can take E and B as the curl of some vector fields. It is convenient to write this as E = ik\1 i\ (vz)

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and B = -ik\1 i\ (uz)

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+ \1 i\ 8,

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(9.69)

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where u and v are two scalar fields and and 8 are two vector fields. When vanishes, E is entirely in the xy-plane. When 8 vanishes, B is also entirely in the xy-plane. Moreover, notice that there is, of course, a gauge freedom: We can add a gradient of an arbitrary function of position to ikvz+ and another to -ikuz+ 8, without changing E and B. Substituting the first equation of (9.69) in (9.64), we find that

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(9.70)

COMPLETENESS IN GENERAL

This means that B - k 2 V Z + ik e is the gradient of some function of position. The gauge freedom in ikvz+ e amounts to this function being arbitrary. Then, substituting the right-hand side of the second equation in (9.69) for B, we obtain

ike - k 2 vz + '\1</; - ik'\1/\ (uz)

+ '\1/\ B = 0,

(9.71)

where </; is an arbitrary function of position. Applying the same reasoning to (9.66), we find that -ikB - k 2 uz + '\1</;' + '\1/\ e + ik'\1/\ (vz) = 0, (9.72) where </;' is another arbitrary scalar function. Thus, we can choose </; and </;' such that