A symmetry of a dynamical system is a diffeomorphism of the configuration space which sends solutions of the equations of motion to solutions of the equations of motion. That is, $A$ is a symmetry if $$x(t)~\text{is a physical evolution of the system}~~~\to~~~A (x(t))~\text{is a physical evolution of the system}.$$ I am being pedantic here, because, often, the word symmetry is also used to describe an operator $A$ which relates the dynamics of two different dynamical systems, sharing the same configuration space: $$x(t)~\text{is a physical evolution of system 1}~~~\to~~~A (x(t))~\text{is a physical evolution of system 2}.$$ I will call the symmetries mentioned first "symmetries of the first kind", and the latter "symmetries of the second kind". I will now give examples:
- For a particle in an n-dimensional potential, all coordinate transformations in $GL(n)$ induce symmetries of the first kind on the phase space.
- For electromagnetism, "local $U(1)$-symmetry" is a symmetry of the second kind, and it refers to conjugation by a unitary operator (a gauge transformation). Gauge-equivalences refer to dynamics being related by a gauge transformation.
Now that I've clearly defined these two, I now claim that symmetries of the second kind are vacuous. Therefore, statements of the type $$\textit{"EM has a local $U(1)$-symmetry"}$$ are thus vacuous. The problem is, they're just referring to changing coordinates on a geometric object. For example, I could extend the notion of a gauge transformation by allowing it to translate the fibres of the respective bundles, or, even worse, let it be discontinuous. Why not call a gauge-transformation, followed by a translation and dilation, a "fundamental symmetry of electromagnetism"? Why not call a hideous symplectomorphism of a phase space a "symmetry", because it happens to be some arbitrary distortion of the dynamics?
