46.5.3 Rotations
We will not go through the exercise of constructing the angular momentum
operator̂Jfor the electromagnetic field that gives the action of rotations on the
theory. Details of how to do this can for instance be found in chapters 6 and
7 of [35]. The situation is similar to that of the spin^12 Pauli-equation case in
section 34.2. There we found thatJ=L+Swhere the first term is the “orbital”
angular momentum, due to the action of rotations on space, whileSwas the
infinitesimal counterpart of theSU(2) action on two-component spinors.
Much the same thing happens in this case, withLdue to the action on spatial
coordinatesx, andSdue to the action ofSO(3) rotations on the 3 components
of the vectorA(see equation 46.19). We have seen that in the Coulomb gauge,
the fieldÂ(x) decomposes into two copies of fields behaving much like the scalar
Klein-Gordon theory, corresponding to the two basis vectors 1 (p), 2 (p) of the
plane perpendicular to the vectorp. The subgroupSO(2)⊂SO(3) of rotations
about the axispacts on this plane and its basis vectors in exactly the same
way as the internalSO(2) symmetry acted on pairs of real Klein-Gordon fields
(see section 44.1.1). In that case the sameSO(2) acts in the same way at each
point in space-time, whereas here thisSO(2)⊂SO(3) varies depending on the
momentum vector.
In the internal symmetry case, we found an operatorQ̂with integer eigen-
values (the charge). The analogous operator in this case is the helicity operator.
The massless Poincar ́e group representations described in section 42.3.5 are the
ones that occur here, for the case of helicity±1. Just as in the internal symmetry
case, where complexification allowed diagonalization ofQ̂on the single-particle
space, getting charges±1, here complexification of the 1 (p), 2 (p) diagonalizes
the helicity, getting so-called “left circularly polarized” and “right circularly
polarized” photon states.
46.6 Covariant gauge quantization
The methods used so far to handle gauge invariance suffer from various problems
that can make their use awkward, most obviously the problem that they break
Lorentz invariance by imposing non-Lorentz invariant conditions (A 0 = 0,∇·
A= 0). Lorentz invariance can be maintained by use of a Lorentz invariant
gauge condition, for example (note that the name is not a typo):
Definition(Lorenz gauge).A vector potentialAμis said to be in Lorenz gauge
if
χ(A)≡−∂A 0
∂t+∇·A= 0
Besides Lorentz invariance (Lorentz transforms of vector potentials in Lorenz
gauge remain in Lorenz gauge), this gauge has the attractive feature that