Quote by atyy
Whereas with finite number of particles, the Hilbert spaces spanned by the different Fock spaces are the same? Or do they get more and more orthogonal with increasing numbers of particles?

Yes, a finite number of particles in an infinite volume corresponds to density 0 and, in the free particle case, to mu=0. For a finite volume, all states with different particle number lie in the same Hilbert space. So to say the chemical potential is a new classical variable which enumerates inequivalent Hilbert spaces in the thermodynamic limit.