Question

I view gauge theory as asking what local symmetries leave our theory invariant and then seeing the consequences. Thus, taking a look at the Lagrangian for electromagnetism, we can act on each point by a U(1) action, i.e. multiply by e^iθ(x), and this would not change the equations of motion. The generalizations of this idea to other symmetry groups and other field theories also make sense to me.

What is confusing me is reconciling this local symmetry idea with how gauge theory is presented in classical electromagnetism books. For example, in E&M you may modify a vector potential A and scalar potential V by the following gauge transformation:
A→A+∇λ, V→V−∂λ∂t (1)
where λ is any scalar function and this has no effect on the physical theory. We can then use the above transformations to conveniently fix a gauge such as the Coulomb gauge or the Lorenz gauge.

I am aware that this can be made rigorous by seeing how the local sections of a principal bundle over spacetime change when acted on by the transition functions of the bundle. But on a more intuitive or physical level, what is the relationship between the above gauge transformations and the idea of a symmetry at every point in space-time?

For example, I don't see what the U(1) local symmetry of the Maxwell theory has anything to do with the gauge transformations (1) or specific gauges such as the Coulomb or Lorenz gauge (or more generally, what it has to do with gauge fixing at all). What does the function λ have to do with the value we choose for θ(x) in the U(1) action?

The two seem like different ideas that I'm sure are related but I cannot see the connection. How can one tie these ideas together?

Answer 1

In gauge theory, the U(1) local symmetry of Maxwell's theory is related to gauge transformations in classical electromagnetism, as both ensure the invariance of the physical laws under certain transformations. The function λ in the gauge transformation serves a similar purpose as the phase θ(x) in the U(1) action, reflecting symmetry in electromagnetism. This highlights the symmetry Maxwell introduced into his equations, emphasizing the consistency of physical laws from different perspectives.

Step-by-step explanation:

The relationship between the local symmetries of gauge theory and the gauge transformations used in classical electromagnetism is deeply interconnected. The U(1) action you mention, multiplying by eiθ(x), is a representation of the local symmetry that does not alter the physics of the theory. This local symmetry, when expressed in terms of the electromagnetic potentials, translates to the transformation A → A + ∇λ, V → V - &partial;λ/&partial;t. Both forms are manifestations of the same principle: that physical laws should be invariant under certain transformations. Gauge fixing, such as in the Coulomb or Lorenz gauge, utilizes these transformations to simplify calculations without changing the physical content loaded by theories. The function λ is related to the phase θ(x) in the sense that both adjust the potentials or fields in a way that is physically inconsequential due to the redundancies encoded in how we describe electromagnetic fields. The coherence between these concepts is exemplified by the fact that Maxwell's equations, encompassing the major laws of electricity and magnetism, remain invariant under these transformations, reflecting a symmetry in physical laws as perceived by different observers.