#### Spinnor

Gold Member

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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!

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!

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