I've seen electromagnetics formulated in the potentials ($\bf A$, $\varphi$) (and their magnetic counterparts) in the Lorenz gauge for a long time. Justifications for using these include that they are less singular than the fields ($\bf E$, $\bf H$) and reduce the redundancy of Maxwell's equations. All the fields derive from two vector and two scalar potentials (total of 8 potentials). In older engineering works (1950s-1969), I've found they don't use the ($\bf A$, $\varphi$) potentials at all, and instead use a set of two vector potentials called the Hertz vectors by the authors, usually noted by ${\bf\Pi}$, from which all the fields derive. They seem to have an additional level of regularity to them because the ($\bf A$, $\varphi$) potentials derive from them. There are also a total of only 6 scalar quantities between the two, from which all the fields derive.
Modern authors seem to prefer the dyadic Green's functions. I haven't studied these in any depth, but it appears to me that in this formulation, all the fields derive from 5 scalar quantities? Not sure about this one so feel free to correct me. But it got me thinking about my questions:
Is 3D classical EM, how many fundamental constraints are there in Maxwell's equations? What is the minimum number of quantities that can be defined, from which all the fields can be said to derive? I suspect the differential forms / tensor formulation will be necessary to get to the bottom of this, but if there is an interpretation in terms of 3D vector calculus, that'd be great.
Best Answer
An electromagnetic wave with a well-defined frequency and direction, i.e. $\vec k$, only has two possible truly physical i.e. transverse polarizations, i.e. the linearly polarized waves in the $x$ and $y$ direction (or the two circularly polarized ones). That implies that a truly physical counting of polarizations gives you 2, more generally $D-2$ in $D$ spacetime dimensions.
Starting from the $A_\mu$ potential fields, one component is unphysical because it's pure gauge, $A_\mu\sim\partial_\mu\lambda$, and one of them is forbidden due to Gauss' constraint $\rm div\,\vec D=0$ etc. that already constrains the allowed initial state of the electromagnetic field. Both of these killed polarizations are ultimately linked to the $U(1)$ gauge symmetry.
If one is allowed to count off-shell and unphysical fields, there may be many more components than two. But it's always possible to deduce that there are two physical polarizations at the end. For example, when we view $\vec B,\vec E$ as basic fields, there are six components, a lot. But these fields only enter Maxwell's equations through first derivatives, and not second as expected for "normal" bosonic fields, so these fields are simultaneously the canonical momenta for themselves. This brings us to three polarizations but one of them is killed by the constraints, the Maxwell's equations that don't contain time derivatives.
The Hertz vector is just the most famous "non-standard" example how to write the electromagnetic field as a combination of derivatives of some other fields. One must understand that the room for mathematical redefinitions etc. is unlimited and it is a matter of pure maths. All these descriptions may describe the same physics. At the end, the only "truly invariant because measurable" number of "fields" that all these approaches must agree about is the number of linearly independent physical polarizations of a wave/photon with a given $\vec k$.
If you can analyze any mathematical formulation of electromagnetism or another field theory and derive that there are $D-2$ physical polarizations (this usually boils down to the difference of the number of a priori fields minus the number of independent constraints and the number of parameters defining identifications i.e. gauge symmetries – but the independence is sometimes hard to see and requires you to make many steps of the counting), then you have proved everything that is "really forced to be true". Various formalisms may offer you other ways to count the number of off-shell fields (with different answers) and they may be useful (because they satisfy certain conditions or enter some laws) but to discuss them, one has to know what the laws where they enter actually are.
A truly physical approach is only one that counts the physical polarizations. The gauge symmetry is just a redundancy, a mathematical trick to get the right theory with 2 physical polarizations out of a greater number of fields with certain extra constraints or identifications. The precise number of constraints or identifications may depend on the chosen mathematical formalism and it is not a physically meaningful question – it is a question of a subjectively preferred mathematical formalism because the physics is equivalent for all of them.