The circle is the fiber over Minkowski spacetime for electromagnetism? I want to make connection to the classical vector potential via some " picture" involving this circle. Does the following come close? Can I consider a 3 dimensional surface in C_1XMinkowski space that at a given slice of time has position in the circle dimension of θ(X,t)? Obviously a different observer would see a Lorentz transformed θ(X,t) --> θ'(X',t') Given the right properties could this surface encode the physics of the classical vector potential, A_μ? I am thinking we could define kinetic energy at X and time t as being proportional to θ(X,t),t? The potential energy would have parts from the divergence of θ(X,t), ∇°θ(X,t) and the curl of θ(X,t), ∇Xθ(X,t)? So The electric field goes as something like ∇°θ(X,t) + θ(X,t),t Edit, we need a vector above and θ(X,t),t is not a vector. Damn! 2nd edit, ∇θ(X,t),t is a vector, does that work? 3rd edit, we can't take the curl of a scaler, ∇Xθ(X,t)? And the magnetic field goes like ∇Xθ(X,t)? How do you simply get electromagnetism from the circle? Mathematics is a very precise language and I apologize for not being more precise. Thanks for any help!