# What's the influence due to the phase factor in state vector?

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.

• What do you mean by "influence"? – ACuriousMind Feb 22 '17 at 9:07
• The phase factor shows the time evolution of phase of the system. It appears even for a stationary state. It doesn't show any influence on the vector – UKH Feb 22 '17 at 9:36
• @ACuriousMind I mean is there any 0bservable effects? – Jack Feb 22 '17 at 9:55
• Everything would be so easy if there was the one-to-one correspondence you are describing. Sadly, there are many very strong suggestions that this should not be the case. The existence of uncountably many inequivalent irreducible representations of the canonical commutation relations for quantum fields is one of such suggestions. Another is the fact that not every quantum state can be represented, in a given (irreducible) representation, as a ray in Hilbert space (or as a density matrix, actually). – yuggib Feb 22 '17 at 10:14

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.