There are transformations on physical states which induce unitary transformations of vectors in Hilbert space that correspond to these physical states. We demand that operators in Hilbert space be unitary and from this we mathematically deduce what should the algebra look like and what form should these transformations have in Hilbert space. These transformations are known as symmetry transformations for they preserve inner product or the norm, which is probability in QM.
So how can we be sure that our reasoning is correct just on these wage and weak arguments? Of course we can derive transformations and of course it all works, but it seems to me that this reasoning lacks something strong. Am I wrong??