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−
∂
∂t
A 0
)
−∇^2 A+
∂^2 A
∂t^2
=−∇^2 A+
∂^2 A
∂t^2
the 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 φ
soφthat 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 ωp
using 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: