W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954), pp. 3–5.

M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), p. 23.

W. D. Montgomery, J. Opt. Soc. Am. 58, 1112 (1968).

The emphasis in the present paper is on the vector potential A rather than on E, as in (I). The reason for this is that the potential enters directly into the interaction hamiltonian between radiation and electron, as well as the fact that the total electromagnetic energy is simply written in terms of the vector potential. See P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1958), pp. 239–241.

R. K. Luneburg, Mathematical Theory of Optics (Univ. of Calif. Press, Berkeley, 1964), pp. 311–319.

N. Dunford and J. Schwartz, Linear Operators Part I: General Theory (Wiley-Interscience, Inc., New York, 1967), pp. 119–121, 241.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press Ltd., London, 1966), p. 33.

This equivalence has been proven earlier by C. J. Bouwkamp, Rept. Progr. Phys. 17, 39 (1954) and G. C. Sherman, J. Opt. Soc. Am. 57, 546 (1967) in mathematical contexts having less algebraic structure. Part of the equivalence proof in (I) was later repeated by E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).

We will term the behavior of D_{z} on b as that of a homogeneous operator, because it acts on each component separately.

Reference 2, pp. 29–32.

Note that *D*_{z}*, as a homogeneous operator in *H*, is still the adjoint of *D*_{z} in *H*.

Equation (16) is implicit in Eq. 2.107 of Clemmow (Ref. 7), p. 33.

A subset *m* of the Hilbert space *H* is a subspace if it is both a linear manifold and closed. A linear manifold *M* is one for which f, g∈*M* implies αf +βg∈*M* for all complex numbers α, β. It is closed (in the norm) if for every sequence {f_{n}}⊆ *M* such that ‖f_{n} — f‖ → 0 as *n* → ∞ for some f∈*H* we must have f∈*M*.

P. R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1957), p. 22.

R. Mittra and P. Ransom in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, New York, 1967), pp. 619–647.

G. C. Sherman, J. Opt. Soc. Am. 57, 1490 (1967).

E. Wolf and J. Shewell, Phys. Letters 25A, 417 (1967).

E. Lalor, J. Math. Phys. 9, 12, 2001 (1968).

J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).

W. D. Montgomery, J. Opt. Soc. Am. 59, 136 (1969).

W. D. Montgomery, J. Opt. Soc. Am. 58, 720A (1968).

G. C. Sherman, Phys. Rev. Letters 21, 761 (1968).

W. D. Montgomery, Nuovo Cimento LIX, 137 (1969).

Reference 2, pp. 104, 105.

Reference 14, pp. 24, 25.

Reference 6, p. 190.