Take 3 dimensions of Space and one of time, you get 4D Spacetime. Mass distorts this field and you get gravity…this is Einstein’s Theory of General Relativity.
Extend the theory to 5 dimensions and you get Gravity AND Electromagnetism…this is Kaluza-Klein.
Now, I has been explained on these boards that GR cannot be quantized, as meaningless infinities arise which do not match what is observed.
What If we limit the quantizing strictly to the electromagnetism aspect of the theory? Would we still get nonsense, or can we derive Quantum Electro Dynamics from it?
You wouldn’t be asking us to do your homework for you, would you?
Kaluza-Klein, AFAIK, works only in the classical limit. The moment you start talking about QED you’re entering into a realm that cannot be described (yet) using tensor calculus. One of the biggest problems I can think of is that QED allows us to combine electromagnetism and the weak force into the electroweak force. I don’t see how such a thing could be done using Kaluza-Klein which only has one more dimension than normal GR.
The whole point of KK is to unify EM and GR. If you separate out the EM part and quantize it, you are just doing QED. You have given up the unification.
Of course, quantizing KK is just as hard as quantizing GR.