We know that observables correspond to hermitian operators on the Hilbert space of physical states of the system. We also know, via Wigner theorem, that for each symmetry there is a linear unitary operator (or anti-linear and anti-unitary). In the case of a continuous symmetry, that is in the case of a Lie group of symmetries, all elements of the group (the connected part of the group) can be written as the complex exponential of a hermitian operator. This hermitian operator is the generator of the corresponding symmetry. So it seems natural to think that all observables may correspond to the generators of some symmetry, given that it is well known that momentum is the generator of spatial translation and angular momentum is the generator of rotations. Is this true? And if so what is the symmetry that correspond to the position observable in quantum mechanics?