Position operator in momentum space generator
How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? derived the position operator in momentum space using commutators.
I want to extend the analogy of equivalence between position and momemtum space. In position space, we say momentum is the generator of translations. This can potentially suggest the form of the momentum operators. Is there a generator of some kind of infinitesimal symmetric transformation in momentum space that, when repeated several times, gives you position?
In other words, consider the statement "momentum is the generator of translation in position space." I want to know if there is an equivalent statement saying "X is the generator of translation in momentum space" i.e. X is a generator of momentum change. I tentatively say this has something to do with force because in Newtonian physics, force changes momentum. But I also see two reasons to think that this new generator should actually be something to do with position:
1 the position operator in momemtum psace looks very similar to the momentum operator in position space.
2 momentum and position are supposed to be connected deeply because wavenumber k space is the Fourier transform of position. It is supposed to be equally informative to work in both position and momentum space.