Electromagnetic tensor use in QFT

(This question may lose me some physics knowledge points, but please be patient with me, my training is mostly in chemistry)

What is the purpose of using the electromagnetic tensor over simply directly working with electric and magnetic fields? Specifically in the quantum electrodynamic Lagrangian we use the tensor. 1 However, in my work with classical electromagnetism I have found that simply working with vector fields is much more profitable and the EM tensor is usually not needed. (If nothing else vector fields are simply easier to use.)

So in brief, why does no quantum Lagrangian exist in terms of the vector fields we know from classical EM?

• We require a relativistically invariant Lagrangian which is easier to construct in terms of relativistically covariant objects such as $A_\mu, J_\mu, F_{\mu\nu}$ rather than in terms of ${\bf E}, {\bf B}, \rho, {\bf J}$. All you have to look for terms where the Lorentz indices are contracted. If you wish, you can express the Lagrangian in terms of Lorentz noncovariant objects ${\bf E}, {\bf B}, \rho, {\bf J}$. – SRS Jun 28 '20 at 2:52

2 Answers

It does. If you don't want to use $$F_{\mu\nu}$$ you can always write the pure QED Langrangian as

$$\mathcal L = (\vec E^2 - \vec B^2)$$

which is equivalent. However it's not so easy to see that this quantity is Lorentz invariant, while it is easy to see that $$F_{\mu\nu}F^{\mu\nu}$$ is. For the purposes of QED, the use of $$F_{\mu\nu}F^{\mu\nu}$$ is more common than the above (though the above can just as easily be quantized, it's just notation after all) because it is more convenient to work in a way that makes the symmetries of the theory manifest.

This may not be what you're looking for, but electric and magnetic fields are not Lorentz invariant, i.e. well-defined in the special theory of relativity. A point particle will (classically) emit only an electric field if it's at rest, but it will emit both electric and magnetic fields if you boost out of its rest frame. Quantum field theory is designed to be Lorentz invariant, so its fundamental objects should be Lorentz invariant objects like four-vectors (e.g. the electromagnetic four potential) and tensors (e.g. the EM tensor, which just consists of derivatives acting on the four potential). This Lorentz-invariant formulation of EM in terms of four-potentials and tensors is actually entirely classical, as you can see here: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Covariant_formulation_in_vacuum. The quantum part is just interpreting the four-potential as a quantum field operator, rather than a classical field with a definite value.

Actually, it turns out that the four potential is somewhat more fundamental than the EM tensor: certain quantum phenomena, like the Aharonov-Bohm effect, rely on the four potential directly, producing measurable effects even if there are no electric or magnetic forces present.