QuantumPhysics.dvi

19 Quantization of the Free Electro-magnetic Field

We shall begin by presenting a brief review of classical Maxwell theory, and then proceed to the
quantization of the free Maxwell field.

19.1 Classical Maxwell theory

Maxwell’s equations for the electric fieldEand the magnetic fieldB, in the presence of an electric
charge densityρand an electric charge current densityjare usually divided into two groups. The
first group of equations does not involveρandjand is given as follows,

∇×E = −∂tB

∇·B = 0 (19.1)

while the second group depends onρandj, and is given by

∇×B =

1

c^2

∂tE+μ 0 j

∇·E =

1

ε 0

ρ (19.2)

Integrability of the equations in the second group (19.2) requires (local) electric charge conservation,

∇·j+∂tρ= 0 (19.3)

Maxwell’s equations are inconsistent unless this equationholds. Notice thatc^2 ε 0 μ 0 = 1, wherec,ε 0 ,
andμ 0 are respectively the speed of light, the electric permittivity and the magnetic permeability
in vacuum.

The two equations in (19.1) may be completely solved in termsof a scalar potentialA 0 , and a
vector potentialA, as follows,

E = ∇A 0 −∂tA

B = ∇×A (19.4)

Note that the customary electric potential Φ is related toA 0 byA 0 =−Φ. GivenEandB, the
corresponding scalar and vector potentials are not unique,sinceEandBare invariant under the
followinggauge transformations,

A 0 → A′ 0 =A 0 +∂tΛ

A → A′=A+∇Λ (19.5)

where Λ is an arbitrary function oftandx. This arbitrariness allows us to impose one extra scalar
condition on (A 0 ,A), and such a condition is referred to as agauge condition, or agauge choice.