In the fundamental postulate of quantum mechanics (page 18), there exists a one-to-one correspondence like that:
$\text{closed quantum system $\leftrightarrow$ Hilbert space ; quantum state $\leftrightarrow$ a ray in Hilbert space } $
but here the ray can carry some additional phase factor $e^{i\phi}$? What's the observable influence due to the phase factor in state vector? I mean if you take inner product the redundant factor will disappear.
A global phase has absolutely no measurable consequences. Experimental values, such as average values and other moments, do not depends on this overall phase. One often keep track of this phase to simplify calculations. For instance, when rotating a state it is often quite convenient to express the final result without constantly eliminating the overall phase.
On the other hand relative phases matter quite critically as they enter through interference terms.
Note that, when formulating QM in terms of density operator, the overall phase of a state $\vert \psi\rangle$ automatically drops out (at least implicitly) of the expression for the density operator $\hat \rho = \vert\psi\rangle\langle \psi\vert$. This formulation is fully equivalent to the Schr$\ddot{\hbox{o}}$dinger formulation in the case of the so-called pure states.
