What is internal space translation? While I was reading the paper named "Classical time crystals," by A. Shapere and F. Wilczek, I found the following transformation.
$$f(x) \to f(x+e) - \frac{df}{dx}*e$$
It says the transformation is combined internal space-real space translation.
What is internal space translation (I guess it is $f(x) \to f(x+e) - \frac{df}{dx}*e$?) in both mathematical and physical sense?
The paper can be found here. The aforementioned formula is at the first paragragh of second column.
 A: The paper refers to a combined internal space-real space transformation.The first part, $\phi(x) \rightarrow \phi(x+\epsilon)$, is a real space traslation, as it shifts the position coordinate by $\epsilon$ as $x \rightarrow x+\epsilon$.
The second part, $\phi(x) \rightarrow \phi(x) -\epsilon\frac{d\phi}{dx}$, is the internal transformation. This naming comes from the fact that this transformation is acting on the angular variable itself ( or the field in the context of QFT or electromagnetism) and not on the spacetime as the traslation $x\rightarrow x+\epsilon$.
This is a kind of gauge transformations. The easiest to visualize is perhaps the electromagnetic gauge $^1$ where you write the vector potential $\textbf{A}$ as $\textbf{A} + \nabla f$ and the electric potential as $V \rightarrow V - \frac{df}{dt}$ which affect neither any of the "real" degrees of freedom of the system nor the fields ($\textbf{B}$ and $\textbf{E}$) themselves.

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*Classical Electrodynamics (1962). JD Jackson. Section 6.4 .

