## ABSTRACT

In Sec. 12.1 it was realized that the content of the microscopic Maxwell-Lorentz equations in (8.49)-(8.52) is contained in the following inhomogeneous wave equations for the components (µ = 0− 3) of the contravariant four-potential {Aµ(x)}:

∂ν∂ νAµ(x)− ∂µ∂νAν(x) = −µ0Jµ(x), (20.1)

see Eq. (12.6). The form in Eq. (20.1) is gauge invariant, but the wave equations for the different components of the potential are coupled. With the aim of establishing a covariant theory for evanescent fields, the Lorenz condition

∂νA ν(x) = 0 (20.2)

is used as a subsidiary condition. In all Lorenz gauges the inhomogeneous wave equations for the components of {Aµ(x)} decouple; that is,

Aµ(x) = −µ0Jµ(x). (20.3) The complete solution to Eq. (20.3) can be written in the integral form [206, 101, 127]

Aµ(x) = Aµinc(x) + µ0

DR(x− x′)Jµ(x′)d4x′, (20.4)

and the physical interpretation of Eq. (20.4) is the following. An incident (inc) four-potential {Aµinc(x)}, which one often may consider as a prescribed quantity, excites a system of charged particles. The electromagnetic field generated by the nonuniform particle motion together with the incident field results in a selfconsistent current density distribution {Jµ(x)}. The four-potential generated by the infinitesimal current element Jµ(x′)d4x′ spreads out in free space-time in a manner described by the retarded (R) scalar propagator [206, 101]

DR(X) = 1

2π θ ( X0 ) δ ( X2 ) , (20.5)

where X = x−x′. The propagator in Eq. (20.5) is manifest covariant, and the step function θ(X0) = θ(c(t− t′)) ensures that the field emitted from the source point r′ at time t′ reach the point of observation, r at a later time t(> t′). The often used Huygens scalar propagator g(X) is just DR(X) multiplied by the speed of light:

g(X) = cDR(X). (20.6)

It can be shown [127] that g(X) can be rewritten in the form

g(X) ≡ g(R, τ) = 1 4πR

δ

( R

c − τ

) , (20.7)

of

where R = |R| = |r − r′| and τ = t − t′. For our study of evanescent fields it is useful to rewrite Eq. (20.4) in the standard form

Aµ(r, t) = Aµinc(r, t) + µ0

g(R, τ)Jµ(r′, t′)d3r′dt′, (20.8)

and then assume that the components of all vector fields, {Fµ(r, t)}, have the generic form

Fµ(r, t) = Fµ(z;q‖, ω)ei(q‖·r−ωt). (20.9)

Physically, we thus imagine that our particle system possesses infinitesimal translational invariance in time, and in space in the direction given by the (real) vector q‖ = (q‖,x, q‖,y, 0). The form in Eq. (20.9) hence relates to a situation where the incident field is monochromatic (angular frequency: ω) and has plane-wave character perpendicular to the z-direction of the Cartesian coordinate system. The ansatz in Eq. (20.9) allows one to reduce the integral relation in Eq. (20.8) to one-dimensional form. Using the abbreviation

Fµ(z;q‖, ω) ≡ F(z) (20.10)

one thus obtains [129]

Aµ(z) = Aµinc(z) + µ0

g(Z)Jµ(z′)dz′, (20.11)

where Z = z− z′. The scalar Green function (propagator) appearing in Eq. (20.11) is given by [155, 127]

g(Z) = i

⊥|Z|, (20.12)

where

q0⊥ = [(ω

c

)2 − q2‖

, (20.13)

in a generalized sense, is the component of the vacuum field wave vector in the z-direction. When q‖ > q0 ≡ ω/c, q0⊥ becomes imaginary, i.e., q0⊥ = iκ0⊥, with

κ0⊥ = [ q2‖ −

(ω c

)2] 12 (> 0). (20.14)

The associated form of the scalar propagator, viz.,

g(Z) = 1

⊥|Z|, q‖ > q0, (20.15)

shows that the four-potential generated by the sheet current {Jµ(z′)}dz′ (located at z′) decays exponentially away from the sheet plane with a spatial decay constant κ0⊥. Vacuum fields of the type

∼ e−κ0⊥|Z| exp [i (q‖ ·R− ωt)] (20.16) are called evanescent, or inhomogeneous, and it is the wave mechanics and the fieldquantized description related to these unusual vacuum fields which are in focus in this

chapter. Evanescent fields can be generated in a number of ways, but since the generation process is unimportant in the present context, it is sufficient for the following analysis to consider the four-potential created by the prevailing current density distribution. Hence, we start from the integral relation

{Aµ(z)} = µ0 2κ0⊥

⊥|z−z′|{Jµ(z′)}dz′, (20.17)

divided in a given frame into its transverse (T)

AT (z) = µ0 2κ0⊥

⊥|z−z′|JT (z′)dz′, (20.18)

longitudinal (L)

AL(z) = µ0 2κ0⊥

⊥|z−z′|JL(z′)dz′, (20.19)

and scalar (S)

AS(z) ≡ A0(z) = µ0 2κ0⊥

parts. In Eq. (20.20) A0 and J0/c ≡ JS denote the contravariant scalar potential and charge density, respectively.