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Two questions concerning the notion of wave function.

1. Every quantum mechanics book describes the space of wave functions (or more generally of quantum states) of a given system as a Hilbert space. But correct me if I'm wrong QM also says that two elements of such a Hilbert space describe the same state provided they are collinear (i.e. provided that one is the product of the other by any non zero complex number). I interpret this property as an indication that the difference between such a pair of wave functions is not physical but is merely a modelling artifact. In that condition why do quantum mechanicians keep representing wave functions as elements of vector spaces rather than of projective spaces. There is a small article on wikipedia on projective Hilbert spaces but without any reference. What is the real value of the vector space structure if it introduces non-physical (and then useless) information?

2. Let A and B be two Galilean frames of reference with respective points of origin [itex]O_A[/itex] and [itex]O_B[/itex] (supposed equal at t = 0) and constant relative vector speed [itex]\overrightarrow{v_{BA}}[/itex]. Let us consider a particle the wave function of which is [itex]\phi_A[/itex] in frame A and [itex]\phi_B[/itex] in frame B. We have at a given time, for a point Q of space:

[itex]\phi_B(Q) = \exp\left (\frac{2 \pi \imath}{h} m \left(\overrightarrow{O_BQ} - \frac{\overrightarrow{O_BO_A}}{2}\right) \cdot \overrightarrow{v_{BA}}\right ) \phi_A(Q)[/itex]

This means the usual notion of wavefunction is an entity dependent on the frame of reference.

Just as a tensor can be described either as a table of numbers with respect to a certain base or more abstractly as a multilinear object independent of bases, has anyone tried to use some more abstract mathematical object for wavefunction such that its projection/coordinates with respect to a certain Galilean frame yields the classical wave function ? The presence of the exponential makes me think about the possible use of some finite dimensional algebra but I couldn't find something that fits well.

Thank you