That is a follow-up to this question: Gauge symmetry is not a symmetry?
Ok, gauge symmetry is not a symmetry, but ...
... a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations.
I want some simple and practical example for this.
So If I, say, take a simple $\phi^4$ theory, then I can gauge it by...
... introducing the proper fake degrees of freedom
Can I?
A trivial example:
Take your original field $\phi$ to be a free real scalar field on $\mathbb{R}^n$. Double the number of fields by adding another free real scalar field $\chi$ to your list of fields
Now introduce a gauge symmetry by making the group of functions $g: \mathbb{R}^n \to \mathbb{R}$ act by $g: (\phi,\chi) \mapsto (\phi,\chi+ g)$. So your group of gauge transformations acts trivially on the space of $\phi$'s and freely and transitively on the space of $\chi$'s.
Now fix a gauge in your favorite way. You can grind through the BRST machinery, or you can just choose the gauge slice $\chi= 0$. Either way your original free scalar field is precisely equivalent to the new theory with two fields and the wacky gauge symmetry I described above.
