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## Main Question or Discussion Point

Hello everybody

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

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