In the Hamiltonian formalism, the momentum Πicanonically conjugate toAiis given by
Πi=∂L
∂A ̇i=ε 0 A ̇i=−ε 0 Ei (19.14)Now, in general, it is an enormous nuisance to keep the quantitiesε 0 ,μ 0 , and evencexplicit. Thus,
henceforth, we shall setε 0 =μ 0 =c= 1. Using dimensional analysis, these constants may always
be restored in a unique way.
19.2 Fourrier modes and radiation oscillators
We begin by considering and solving Maxwell’s equations without sources, namely forρ=j= 0,
in which case we are left to solve the following equations,
∂^2 tA−∆A = 0
∇·A = 0 (19.15)Both are linear partial differential equations with constantcoefficients, and may be solved using
Fourrier analysis. It will be convenient to Fourrier transform only in space, but not in time; thus
we define this Fourier transformC(t,k) by,
A(t,x) =∫ d (^3) k
(2π)^3
C(t,k)eik·x (19.16)
so that (19.15) reduce to,
(∂t^2 +k^2 )C(t,k) = 0
k·C(t,k) = 0 (19.17)
The Fourrier transformC(t,k) is a complex function. Thus, the reality of the fieldArequires a
complex conjugation relation onC,
C(t,−k)∗=C(t,k) (19.18)
The last equation of (19.17) implies that for any givenk, the Fourrier transformC(t,k) is perpen-
dicular, ortransverse, to the momentumk, whence the name for this gauge condition. Thus, for
givenk,C(t,k) takes values in the 2-dimensional space orthogonal tok. It will be convenient to
choose a basis of two orthonormal vectorsεα(k) for this transverse space,
k·εα(k) = 0
ε∗α(k)·εβ(k) = δα,β α,β= 1, 2 (19.19)
The first equation of (19.17) onCis a harmonic oscillator with frequency
ω=ωk=|k| (19.20)