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

Question

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?

Answer 1

You probably never "understood" this because there is not much to understand. There is no single sentence one could point to that directly connects the mercilessly abstract view of gauge theories as the theories of connections on principal bundles on one end and the electric and magnetic fields we know. These two things are at the two extreme ends of a sliding scale of abstraction, and to understand their connection you need to understand all the steps along the way. To understand the quantization of gauge theories, you must make sure that you are already comfortable with the quantization of "ordinary" field theories. If you're just "new to quantum field theory", just be patient. Learn to walk (quantize a scalar field, quantize a Dirac spinor, have some interactions) first before you try to run head-first into the quantization of gauge theories. Here are some remarks, however:

First, what is written in terms of creation and annihilation operators in quantum field theory are the free fields, not the interacting fields that actually determine what happens. You cannot see a connection between that free field and any force because it doesn't describe (the potential of) any actual force, it is a convenient fiction to define a particle interpretation in the field theory.

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.

The other two fundamental forces whose quantum theory is known are now described by non-Abelian gauge groups, but exactly the same Lagrangians - the standard kinetic term for a gauge field together with the gauge field itself coupling to a current charged under it. However, due to their absence at more classical scales - for the strong force, due to confinement, for the weak force, due to the high mass of its bosons - we cannot easily examine their classical limit to reassure us the quantization of such non-Abelian gauge theories is meaningful. Nevertheless, it turns out that the predictions fit rather well with what we observe, mostly in colliders since the low energy regime is difficult to access in QCD.

It might be worthwhile to say that the relevance of the bundle view of gauge theory resurfaces in particular when considering so-called anomalies, which usually are obstructions to proper quantization that are non-perturbative and global in nature. However, the BRST quantization procedure itself is probably more naturally seen in the Hamiltonian view of gauge theory, which is as certain types of constrained systems. For details on these topics, see for instance Bertlmann's Anomalies and Henneaux and Teitelboim's Quantization of Gauge Systems, although the latter doesn't mention the bundle viewpoint of gauge theories.