Transition amplitude vs. transition probability In quantum mechanics, a physical system corresponds to a Hilbert space $\mathscr{H}$. States correspond (not in a one-to-one way) to points in $\mathscr{H}$ and the physical postulate is that the "transition amplitude" (a complex number) from a state corresponding to $v$ into a state corresponding to $u$ is given by: $$ \mathscr{A}(v\to u) = \frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}$$
Where the physical meaning of the transition amplitude is that if you take the squared absolute value of this complex number, you get the actual probability of the system going from the state corresponding to $v$ into the state corresponding to $u$.
My question is: is there any physical information contained in the phase of $\mathscr{A}(v\to u)$? 
If yes:
1) What is this meaning and how do we measure it in the laboratory?
2) When working with projective Hilbert spaces, this information (AFAIK) has to be discarded, because the transition amplitude between two rays (a ray is defined as an equivalence class of $\mathscr{H}$: the equivalence class corresponding to $v\in\mathscr{H}-\{0\}$ is given by $\left[v\right]\equiv\{u\in\mathscr{H}|\exists\lambda\in\mathbb{C}:u=\lambda v\}$) is defined as (for example Bargmann 1964):  $$ \mathscr{A}(\left[v\right]\to \left[u\right]) = \left|\frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}\right|$$ and this seems like a sensible definition because otherwise, setting $$ \mathscr{A}(\left[v\right]\to \left[u\right]) = \frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}$$ results in a not well-defined map $\mathscr{A}$.
Thus, if indeed this information is discarded when working with projective Hilbert spaces, why is it allowed to discard it if it contains physical information?
 A: Well, for an individual measurement, for example in the double slit experiment with a single electron, the phases are what make the interference pattern in the accumulation, so in this sense they have measurable effects.
In the superposition of photons to make up the classical electromagnetic wave, the phases are also important. This experiment which shows interference of two independent laser beams relies on the superposition of the amplitudes and the phase information is important. 
A: It sounds to me like you are asking for the difference between relative and absolute phase.  It is true that there is never a meaningful absolute phase of any complete state of an isolated system, for the reason you mention-- multiplying that state by any complex number has no effect on any observable.  Also note that since your approach to defining states does not require that they be normalized, it's not just the absolute phase, but also the absolute magnitude, that has no physical significance.  In a geometric picture, all that physically matters is the direction in which the state "points."
However, an important rule about states is that you can add superpositions of them to get a new state.  Neither the phase nor magnitude of the new state matters either, but what does matter is the relative phase and magnitude of the states you are superimposing to get the new state, because they will affect the "direction" of the superimposed state.  
So that suggests pretty strongly to me that the concepts of phase and magnitude, in the context of the physical information they represent, only have meaning in regard to the adding of states, i.e., when generating superpositions.  Hence, the meaning of phase and magnitude are inextricably connected to the meaning of a superposition, and that's how they connect to the concept of interference in the two-slit experiment and so on.
