Psy-Kosh: I think that is a great question. Here is my take on it:
The wavefunction for six particles will be a function of six variables, x1,y1,z1,x2,y2,z2. You could of course think of these as just six variables without thinking in terms of two particles with three coordinates apiece. However, from this point of view, the system would have certain strange properties that appear coincidental. For example, suppose the two particles are bosons. Then, if we exchange them, nothing happens to the wavefunction. This seems fairly natural. However, from the 6D point of view we have the strange property that if we swap three particular pairs of variables (x1 swapped with x2, y1 swapped with y2, and z1 swapped with z2) the wavefunction is unchanged, whereas in general if we pair the variables in any other way and swap them the wavefunction is changed. Similarly, the potential term in the Hamiltonian will often depend on the distance between the two particles (such as if they repel coulombically). This again seems natural. However, from the 6D point of view this is a mysterious property that the potential depends only on (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2, where we have subtracted variables in pairs in some particular way, rather than in any of the many other ways we could pair them.
Psy-Kosh: I think that is a great question. Here is my take on it:
The wavefunction for six particles will be a function of six variables, x1,y1,z1,x2,y2,z2. You could of course think of these as just six variables without thinking in terms of two particles with three coordinates apiece. However, from this point of view, the system would have certain strange properties that appear coincidental. For example, suppose the two particles are bosons. Then, if we exchange them, nothing happens to the wavefunction. This seems fairly natural. However, from the 6D point of view we have the strange property that if we swap three particular pairs of variables (x1 swapped with x2, y1 swapped with y2, and z1 swapped with z2) the wavefunction is unchanged, whereas in general if we pair the variables in any other way and swap them the wavefunction is changed. Similarly, the potential term in the Hamiltonian will often depend on the distance between the two particles (such as if they repel coulombically). This again seems natural. However, from the 6D point of view this is a mysterious property that the potential depends only on (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2, where we have subtracted variables in pairs in some particular way, rather than in any of the many other ways we could pair them.