- The Lorenz gauge condition is needed to get Maxwell’s equations, but it
cannot be imposed as an operator condition
−
∂ 0
∂t+∇·Â= 0
since this is inconsistent with the canonical commutation relation, because
[
 0 (x),−∂ 0 (x′)
∂t+∇·Â(x′)]
=
[
 0 (x),−∂ 0 (x′)
∂t]
=iδ(x−x′) 6 = 0- The commutation relations 46.21 for the operatorsa 0 (p),a† 0 (p) have the
wrong sign. Recall from the discussion in section 26.3 that the positive
sign is required in order for Bargmann-Fock quantization to give a unitary
representation on a harmonic oscillator state space, withaan annihilation
operator, anda†a creation operator.
We saw in section 46.3 that inA 0 = 0 gauge, there was an analog of the
first of these problems, with∇·Ê= 0 an inconsistent operator equation. There
∇·Eplayed the role of a moment map for the group of time-independent gauge
transformations. One can show that similarly,χ(A) plays the role of a moment
map for the group of gauge transformationsφsatisfying the wave equation
46.20. In theA 0 = 0 gauge case, we saw that∇·Ê= 0 could be treated not
as an operator equation, but as a condition on states, determining the physical
state spaceHphys⊂H.
This will not work for the Lorenz gauge condition, since it can be shown
that there will be no states such that
(
−
∂ 0
∂t+∇·Â
)
|ψ〉= 0The problem is that, unlike in the Gauss’s law case, the complex structureJr
used for quantization (+ion the positive energy single-particle states,−ion
the negative energy ones) does not commute with the Lorenz gauge condition.
The gauge condition needs to be implemented not on the dual phase spaceM
(here the space ofAμsatisfying the massless wave equation), but onH 1 =M+Jr,
where
M⊗C=M+Jr⊕M−Jr
is the decomposition of the complexification ofH 1 into negative and positive
energy subspaces. The condition we want is thus
χ(A)+= 0whereχ(A)+is the positive energy part of the decomposition ofχ(A) into
positive and negative energy components.
This sort of gauge condition can be implemented either before or after quan-
tization, as follows: