Quantum mechanics and Lorentz symmetry The operator $P$ in quantum mechanics is the generator for the translation transformation. We have:
$$\exp(iPa)|x\rangle=|x+a\rangle$$
Similarly, I think the operator $X$ is the generator for the Galileo transformation:
$$\exp(-iXq)|p\rangle=|p+q\rangle$$
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?)
 A: In non relativistic quantum mechanics, referring to an irreducible projective unitary representation of Galileo group, up to a multiplicative factor (the mass) the position operator naturally arises as the generator of boost transformation. This is equivalent to the standard translation in the space of momenta.
In relativistic QM  the standard translation in the space of momenta ceases to be a symmetry and the generator of the boost has another form.
The relativistic definition of the position operator is more involved. It is possible but uses a different approach, technically based on the so called imprimitivity structures. It is possible to prove that, for elementary systems (unitary irreducible representations of Poincaré group), the position observable is uniquely defined for massive systems, otherwise it is not always well-defined, depending on the value of the spin. The position operator is also known as Newton-Wigner position operator.
A good reference is the book by Varadarajan Geometry of Quantum Theory, in one of the last chapters it studies the problem in details. Also in  Barut Raczka's textbook Theory of group representations and applications there is a detailed, but less rigorous, discussion in one of the last chapters.
