46.5 Space-time symmetries
The choice of Coulomb gauge nicely isolates the two physical degrees of freedom
that describe photons and allows a straightforward quantization in terms of two
copies of the previously studied relativistic scalar field. It does however do this in
a way which makes some aspects of the Poincar ́e group action on the theory hard
to understand, in particular the action of boost transformations. Our choice of
continuous basis elements for the space of solutions of the Maxwell equations
was not invariant under boost transformations (since it uses initial data at a
fixed time, see equation 46.5), but making the gauge choiceA 0 = 0,∇·A= 0
creates another fundamental problem. Acting by a boost on a solution in this
gauge will typically take it to a solution no longer satisfying the gauge condition.
For some indication of the difficulties introduced by the non-Lorentz invari-
ant Coulomb gauge choice, the field commutators can be computed, with the
result
[Âj(x),Âk(x′)] = [Êj(x),Êk(x′)] = 0
[Âj(x),Êk(x′)] =−
i
(2π)^3∫
R^3(
δjk−
pjpk
|p|^2)
eip·(x−x′)
d^3 p (46.18)The right-hand side in the last case is not just the expected delta-function, but
includes a term that is non-local in position space.
In section 46.6 we will discuss what happens with a Lorentz invariant gauge
choice, but for now will just consider the Poincar ́e subgroup of space-time trans-
lations and spatial rotations, which do preserve the Coulomb gauge choice. Such
group elements can be labeled by (a,R), wherea= (a 0 ,a) is a translation in
space-time, andR∈SO(3) is a spatial rotation. Generalizing the scalar field
case (equation 44.6), we want to construct a unitary representation of the group
of such elements by operatorsU(a,R) on the state space, with theU(a,R) also
acting as intertwining operators on the field operators, by:
Â(t,x)→U(a,R)Â(t,x)U−^1 (a,R) =R−^1 Â(t+a 0 ,Rx+a) (46.19)To constructU(a,R) we will proceed as for the scalar field case (see sec-
tion 44.2) to identify the Lie algebra representation operators that satisfy the
needed commutation relations, skipping some details (these can be found in
most quantum field theory textbooks).
46.5.1 Time translations
For time translations, as usual one just needs to find the Hamiltonian operator
Ĥ, and then
U(a 0 , 1 ) =eia^0
Ĥ