# How to understand the connection between the fundamental forces and gauge groups?

I'm new to Quantum Field theory and that's something I've always seem people talk about but I've never understood.

On one hand we have the forces themselves. As I understood, the description within the framework of Quantum Field Theory is that each force is described by a Quantum Field. I've seem this done on Merzbacher's book on Quantum Mechanics, where in the last chapter he deals with Electromagnetic Fields and Photons.

The Quantum Field then is written in terms of creation and annihilation operators.

On the other hand, we have the theory of connections on principal bundles. The idea of Gauge Group fits into this mathematical framework. Indeed, if we have a principal bundle $\pi : P\to M$ with structure group $G$, we know that each fiber $\pi^{-1}(q)$ for $q\in M$ is isomorphic to $G$ and, furthermore, a choice of gauge is just a local trivialization $\sigma: M\to P$. The group $G$ is then the gauge group.

Now, I just can't understand how these two apparently totally different things relate.

So, just for the sake of an example, electromagnetic force is usually associated to the group $U(1)$.

We can indeed, pick any base space $M$ and consider a principal bundle $\pi : P\to M$ with structure group $U(1)$. But what this has to do with the Quantum Field Theory version of electromagnetism?

How to understand, in general, this connection established between the fundamental forces, described as Quantum Fields, and gauge groups?

Standard quantum field theory proceeds by taking a classical field theory Lagrangian and applying quantization prescriptions (usually the canonical or the path integral) to it. The Lagrangian formulation of electromagnetism is well-known to be that of a $\mathrm{U}(1)$ gauge theory, with the electromagnetic four-potential as the gauge connection. The base space is of course spacetime, the $\mathrm{U}(1)$-principal bundle is usually the trivial one (there are no non-trivial $\mathrm{U}(1)$ bundles over Minkowski space). Quantizing this theory - with particular attention to the nature of gauge invariance, called Gupta-Bleuler quantization in this Abelian case and BRST quantization in the case of a general gauge group - yields quantum electrodynamics, from which one, in turn and reassuringly, may derive the classical Coulomb potential and hence the classical force.