(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?