## ABSTRACT

Let us now return to the plane-mode decomposition of the covariant four-potential already discussed in Subsec. 28.2.2 and and Sec. 28.3, and let the spatial mode spectrum be discrete. It appears from Eqs. (28.57) and (28.63) that the classical four-potential expanded after the complete set of orthonormalized fr(x,q) and f

∗ r (x,q) mode functions now reads

A(x) ≡ {Aµ(x)} = ∑ r,q

[αr(q)fr(x,q) + α ∗ r(q)f

≡ ∑ i

[αifi(x) + α ∗ i f ∗ i (x)] . (29.1)

In the last member of Eq. (29.1) the abbreviated notation i = (r,q) has been used. The mode functions satisfy a generalized orthonormalization condition, viz.,

〈fi|fj〉 = ζiδij , (29.2) 〈f∗i |f∗j 〉 = −ζiδij , (29.3) 〈fi|f∗j 〉 = 0, (29.4)

where, for all q, ζi = +1 for r = 1− 3, and ζi = −1 for r = 0. In a mode expansion of A(x) the choice of mode functions is not unique. Hence, one may consider the expansion in Eq. (29.1) in a generalized sense: There exists a complete set of positive [fi(x)]- and negative [f∗i (x)]-frequency mode solutions to the free-space wave equation ∂µ∂

µA(x) = 0, and the modes are orthonormal in the relativistic scalar product [Eq. (28.13), with K given by Eq. (28.18)].