So it means whether there is phonon BEC depends on how you quantize the system? But does it have any physical implications whether there is BEC or not? Is the ambiguity a particular trait of phonons (quasiparticles)? Rubidium atoms have BEC regardless of how you describe it right?
I notice that some bosons can exhibit Bose-Einstein condensation while others cannot (photons, phonons). Is it true that the bosons can have BEC only when the total number of particles is conserved? In this case, the chemical potential approaches zero at T_c, and particles begin to cluster...