Physics Asked on July 24, 2021
Forgive me if what I’m asking is too naive for this site. I’m a math student who is recently studying electrodynamics and gauge theory myself. While I’m aware of the fact that QED can be realized as a $U(1)$ gauge theory, I don’t know if I have a valid understanding of this statement.
Let $A$ denote the electromagnetic 4-potential. I know that under a local gauge transformation $mathcal{A} mapsto mathcal{A} + dLambda$, the wave function transforms correspondingly $Psi’ mapsto e^{frac{iqLambda}{hbar}}Psi$. Meanwhile, the Lagrangian $mathcal{L}_{QED}$ remains unchanged under appropriate transformation of the covariant derivative. In that regard, the Lie group $U(1)$ describes the local symmetry, and the potential $A$ can be understood as a connection on the principle $U(1)$-bundle.
In the geometric realization, one can visualize the $U(1)$-bundle as the spacetime $mathbb{R}^{1+3}$ with circles attached at each point. However, I’m wondering whether there is a physical interpretation of the fibre? It is very attempting to think the fibre as all possible values of $Psi(x,t)$ at a fixed spacetime point under different choices of $A$ (they have the same modulus). However, I have never seen this narrative in any physics book.
You are mostly correct. However, I will note that it is the fields which transform under the gauge transformations, not the states. Indeed, the (physical) states are required to be gauge invariant. This can be understood in terms of demanding observables (expectation values/correlators) to be gauge invariant, but is also equivalent to the following: there are (broadly) two major quantization schemes for gauge theories.
One which first eliminates the constraints implied by the gauge invariance (for Maxwell's equations this would be the Gauss' law on B, but more broadly is the demand that the Biancci identities of the gauge theory are satisfied). In this scheme, states which are not gauge invariant never enter the picture. For a detailed description of this calculation, see volume 1 of Weinberg's QFT series (it's a difficult read, but there are many very good things in it).
The second strategy would be to quantize without worrying about the constraints and then impose the condition that all states are gauge invariant after the fact, essentially building a larger Hilbert space and then restricting to a subspace which respects the gauge invariance. This strategy can be implemented very nicely via the BRST formalism and has close relations to the Faddeev-Popov approach to gauge fixing. This may be of more interest to you if you're looking to think about the theory as a theory on a $U(1)$ principle bundle since the BRST operator used in this approach is essentially the vertical component of the differential over the bundle when we perform the usual decomposition into vertical and horizonal distributions.
There are many beautiful aspects of these things such as a relation between the above demands of gauge invariance and the on-shell vanishing of the Noether charges associated to the gauge transformations, but before getting to the second part of your question, let me clarify one last point on the setup which will lead nicely to the rest of my answer. There are a couple bundles floating around in the geometric picture. We first have the bundle of which the fields $Psi$ are sections. We also have the principle $U(1)$ bundle you mention.
These two bundles are related via the frame field description. I maintain that this picture is a little more natural (though technically more complex) when we look at fields which are sections over the tangent or cotangent bundles, but nonetheless, our field $Psi$ is a section over some $mathbb{C}$ bundle over spacetime in the $U(1)$ case.
Now, we can introduce a frame field $z$ valued in $U(1)$ such that $Psi=psi z$ where $psi$ is some complex-valued function over spacetime. This field $z$ is now a section over the $U(1)$ bundle you are thinking about, and there is a natural group action on $z$ given by the transition functions which induces an action of the transition functions on $psi$ such that $Psi$, which should be independent of the frame fields, remains invariant.
For a book that goes into some of these things (at least to some level), see V. Parameswaran Nair's QFT book. There is also this paper. Though its interest is in anomalies, the introductory sections discuss aspects of the geometric formulation of gauge theories.
Correct answer by Richard Myers on July 24, 2021
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