I've been reading Jackson's Electrodynamics chapter 6.
I want to believe I now understand the fundamental theorem of vector calculus.
A vector field seems to be decomposable into a longitudinal or irrotational part and a transverse or solenoidal part.
When one applies this idea to the current density,
One can arrive at
$\nabla^2 A - \frac{1}{c^2} \frac{\partial^2 A}{\partial t^2} = - \mu J_{t}$ (hence tranverse guage) (6.30)
$J_t = \frac{1}{4 \pi} \nabla \times \nabla \times \int \frac{J}{|x - x'|} d^3 x$ (6.28)
The instantaneous Coulomb potential due to charge density is
$\phi(x,t) = \frac{1}{4 \pi \epsilon_0} \int \frac{rho(x',t)}{|x - x'|} d^3x'$ (6.23)
All the equations are ok.
This gauge is unphysical.
How is this fixed in this baby version of the problem?
I am a little confused because there is potentially related talk in quantum field theory about ward identities and such and guaranteeing covariance (See for example AccidentalFourierTransform's response to a question here) in the coulomb gauge, if someone can connect these that would be nice.
Please also see the section "Lorenz gauge" in this Wikipedia article. It says " To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as Ward identities. Classically, these identities are equivalent to the continuity equation" in a very similar potentially same context.