Maxwell’s equations become just the standard massless wave equation. Since
∇×B−
∂E
∂t=∇×∇×A+
∂^2 A
∂t^2−
∂
∂t∇A 0
=∇(∇·A)−∇^2 A+
∂^2 A
∂t^2−
∂
∂t∇A 0
=∇
(
∇·A−
∂
∂tA 0
)
−∇^2 A+
∂^2 A
∂t^2=−∇^2 A+∂^2 A
∂t^2the Maxwell equation 46.3 is the massless Klein-Gordon equation for the spatial
components ofA.
Similarly,
∇·E=∇
(
−
∂A
∂t+∇A 0
)
=−
∂
∂t∇·A+∇^2 A^20
=−
∂^2 A 0
∂t^2+∇^2 A 0
so Gauss’s law becomes the massless Klein-Gordon equation forA 0.
Like the temporal gauge, the Lorenz gauge does not completely remove the
gauge freedom. Under a gauge transformation
−
∂A 0
∂t+∇·A→−
∂A 0
∂t+∇·A−
∂^2 φ
∂t^2+∇^2 φsoφthat satisfy the wave equation
∂^2 φ
∂t^2=∇^2 φ (46.20)will give gauge transformations that preserve the Lorenz gauge conditionχ(A) =
0.
The four components ofAμcan be treated as four separate solutions of the
massless Klein-Gordon equation, and the theory then quantized in a Lorentz
covariant manner. The field operators will be
Âμ(t,x) =^1
(2π)^3 /^2∫
R^3(aμ(p)e−iωpteip·x+a†μ(p)eiωpte−ip·x)d^3 p
√
2 ωpusing annihilation and creation operators that satisfy
[aμ(p),a†ν(p′)] =±δμνδ(p−p′) (46.21)where±is +1 for spatial coordinates,−1 for the time coordinate.
Two sorts of problems however arise: