The operator $P$ in quantum mechanics is the generator for the translation transformation. We have:
Similarly, I think the operator $X$ is the generator for the Galileo transformation:
Is this ultimately not consistent with the Lorentz transformations? Did we base on Galileo transformations somewhere when we construct quantum mechanics?
EDIT: Sorry I just found out that the summation of momentum has nothing to do with Galileo transformations... Anyway if you can refer me to some more informations or give me some insights about this it would be appreciated. Like:
What is the transformation that the position operator generates?
The Lorentz boost is an unitary transformation. What is the associated oversable? (What happens in case we consider Galileo transformation?)