There is a simple extension to electromagnetism that lead to a connection to quantum mechanics form electromagnetism. Quantum mechanics are essentially just a version of electromagnetism. So what's the connection.

The extension of EM that we will use are that everything is a superposition of current loops that has zero mass and where the velocity is at the speed of light. If everything that we model originates from this object the energy density will be an invariant under Lorenz transformations and hence enables us to add another extension: the energy density has a limit or the electrical potential in a point cannot be infinitely high and the limit is a universal constant at least locally here in time and space. Now we will finally add another rule. two current segment interact only if they are parallel and directed in the same direction. Also we add to this that the interaction is either only magnetic in case we look at it from the lab frame or only electric if we look at it from the rest frame.

The QED has just the electric interaction and produces excellent result, why is there not a magnetic term. Now you know the solution to this mystery.

Also this model can explain why the reduced electron mass improves the energy calculation in QED. It all about a scaling property of energy density and an assumption that the charge is located in two dimensional spherical symmetric structures where one conclude that the proton removes energy density from the electron and hence we can scale that up to the limit assuming the electron's max density are at the limit.

We are talking about spherical structures as we assume that spherically symmetry is a defining properly of matter and also a 2D shelling is the norm with different current streams in them. QED Simply help's setting up the analysis of this and enables quark and other particles to be combined by simply formulate the theories in a way that enables the powerful use of quaternion algebra.

We can show directly how a Hydrogen atom with an non spin electron can be constructed and connected via both the electromagnetic framework and the QED framework and this connection produces very exact energy levels of hydrogen just from assuming non radiation and satisfying the Klein Gordon equation.

You can read the detail in:

The connection in math Klein Gordon

Now how can we go further. And in the "connection in math" article above we simply base our building block on solution to the Dirac equation and then simply note that we must be able to make paths that in 3D are parallel in order to interact and they need to attain maximum energy density. Through these arguments the mass will be a function of charge and the limit of how high the energy density can be. This hole system will be energetically stable and lead to what we call the proton mass. The QM framework is a work house that enables us to setup this complex geometrical constructions and with the help of algebra it's humanly possible to do that. On the other hand this while setup means that the analysis of the atom can be simplified greatly from trying to solve the Dirac equation or Schrodinger equation directly. And combining the intuition from both worlds enables a motivation of Randell Mills approach to GUTCP and a possibility to make his theory stringent enough to merit a wider acceptance. In the end it is not unlikely that we could solve the energy level problem for all atoms and ions with just a smartphone.