A gauge symmetry is a symmetry (often local) which doesn't affect the physical quantities or the Lagrangian.
A translation of the whole Universe 3 meters to the left won't affect anything physically.
So does this mean translation is a gauge symmetry?
If so, how do we fix a gauge in this symmetry?
I imagine a scalar field under a translation in the direction $a^\mu$ has the transformation $\delta \phi(x) = a^\mu\partial_\mu \phi(x)$. Can we fix a gauge in this case?
But if we fixed a gauge does this mean the gauge-fixed theory is no-longer translation invariant?