I have a model of particles, which unifies all basic physical interactions: electrostatic, strong and gravitational. In this model, particles are represented as spheric formation of fundamental field, and we do not need any quarks. I consider particles as made from continued field which can be represented by means of unified "primary "elements". All kinds of Matter are constructred by such elements. We can describe their properties, but we cannot extract and separete them from continued field.
IMO, it is impossible to describe particles using electromagnetics equations, we need equations which describes "primary elements". These New physics's equation will describe both, particles and electromagnetic field.
I'm not that familiar with field theory, as you are. But I'm sure that you are on the right way. You can try to describe equations for "primary elements", if you wish. And we'll get New physics
Why they must slide? Is there some additional force which make them to slide?
Just do not look back to General relativity.