Why isn't light described in terms of the vector potential?

[Phrak]

Why is the light assumed to be harmonics of the electric and magnetic fields rather than harmonics of the vector potential?

Am I missing something? Whenever details about light are given in classical physics we are always told about second derivatives of E and B. Why isn't light given as a second derivative of the vector potential in four dimensions? It's a perfectly good wave equation and propagates are c.

To put everything up front, A^mu is a four dimensional vector. A^mu = (A^t, Aⁱ). It has three spacelike components and one timelike component. It is Lorentz convariant. Why shouldn't light be the second derivatives of A^mu?

We could describe the motion of a pendulum as d⁴x/dt⁴= cos(omega t), but why go to this extent when the zeroth derivative will do.

 

[Dale]

I'm with you, I like the potential formulation better than the field formulation whenever possible.

 

[Meir Achuz]

The two formulations are just two different ways of doing the same thing. Some things are easier in one or the other. For, instance polarization is easier with E.

 

[Mentz114]

The electric and magnetic fields are observables, and I've heard it said A is not. This latter puzzles me because it's possible to write a problem in terms of an electrical potential, A⁰ which certainly is observable. So, what do people mean when they say the 4-vector A is not observable ?

Re, Arahanov-Bohm effect, from Wiki

The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, A, must be used instead

but

In classical electromagnetism the two descriptions were equivalent.

 

[Dickfore]

The Aharonov-Bohm effect has to do with the change in the phase of the wave function of a charged particle when passing near a localized magnetic field. But, the phase of a wave function is not an observable precisely in the same way as the electromagnetic potentials are not observables.

Sure, you may detect interference fringes change when the electromagnet is turned on. But, I claim that this is exactly due to the B-field and not the A-field being present between the slits and the screen. Namely, what will happen to the fringes if the solenoid was not in the way?

 

[Phrak]

In my understanding of the history of electromagnetism, Maxwell believed that the vector potential was a real field and used it often in his writings. It was only later, due in the most part to Hertz and one other--can't recall who, where the present formulas we are familiar with today, in terms of 4 differential equations or 4 integral equations, became standard. On physicist, Sean Carroll refers to the differential equations as the "Maxwell-Hertz equations."

But getting back to a wave equation in the vector potential, I don't believe that wave equations in A and phi(the electric potential) are isomorphic to wave equations in E and B. That is, I think that we can have wave equations in A and phi that do not result in wave equations in E and B. This is a hunch--I haven't proved it.