A question on gauge fixing As I understand it, a physical theory that has a gauge symmetry is simply one that has redundant degrees of freedom in its description, and as such, is invariant under a continuous group of (in general) local transformations, so-called gauge transformations. 
With this in mind, consider electromagnetism as a prototypical example. This has a $U(1)$ gauge symmetry, such that the theory is invariant under transformations of the vector four-potential, $A^{\mu}$ $$A^{\mu}\rightarrow A'^{\mu}=A^{\mu}+\partial^{\mu}\Lambda(x)$$ where $\Lambda(x)$ is some local function of space-time coordinates.
Is it then correct to say that the theory is described by an equivalence class of  vector four-potentials, $A^{\mu}$ such that $$A'^{\mu}\cong A^{\mu}\iff A'^{\mu}=A^{\mu}+\partial^{\mu}\Lambda(x)$$ Given this, does "choosing a gauge " simply amount to choosing a particular four-potential $A^{\mu}$ from this equivalence class?
Furthermore, does "fixing a gauge" simply amount to specifying some constraint on the choice of $A^{\mu}$ such that it "picks out" a single four-potential $A^{\mu}$ from this equivalence class? 
For example, is it correct to say that choosing the Lorenz gauge $\partial_{\mu}A^{\mu}=0$ partially removes gauge freedom, since it restricts ones choice of four-potential $A^{\mu}$ such that it satisfies $\partial_{\mu}A^{\mu}=0$, however, it doesn't fully fix ones choice of $A^{\mu}$, i.e. it doesn't fully "fix the gauge", since there remains a subspace of gauge transformations that preserve this constraint, corresponding to gauge functions $\psi$ that satisfy the wave equation $\partial_{t}^{2}\psi=c^{2}\nabla^{2}\psi$?! 
 A: Yes, all those things are correct.
The equivalence class of potentials that are related by gauge transformation is called a gauge orbit, since it is an orbit for the action of the group of gauge transformations on the space of potentials.
Choosing/fixing a gauge means picking out particular representants $A$ from each gauge orbit according to a rule encoded by $F[A] = 0$ for some functional $F$, i.e. you select those potentials which fulfill the equation. A partial gauge fixing is indeed given by things like $F[A] = \partial_\mu A^\mu$, for which $F[A] = 0$ has more than one solution in a given orbit. These solutions are related by gauge transformations with harmonic parameter functions, and these transformations are called residual gauge symmetry.
In general, it is not possible to fix a gauge that choose only one representant from every orbit. This is known as the Gribov problem.
A: "does "fixing a gauge" simply amount to specifying some constraint on the choice of $A^\mu$ such that it "picks out" a single four-potential $A^\mu$ from this equivalence class?" Generally speaking, not quite. For example, if you consider, say, interacting Maxwell and Dirac fields, you have to change the phase of the Dirac field when you change the gauge  for the Maxwell field to provide gauge invariance.
