# 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, 2020 at 2:52

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.