  In the first part we investigate in detail if the Bose-Einstein condensation
scenario for magnons can quantitatively explain the observed field-induced
magnetic ordering in TlCuCl$_3$. We use a bond-operator approach to map the
spin system onto hard-core bosons and exactly account for the hard-core
constraint in the dilute limit. We solve the hard-core model within the
Hartree-Fock-Popov approximation and discuss its validity and the consequences
of this approximation for the critical properties. In the second part the
effects of spin-phonon and spin-orbit coupling are discussed within this
framework. We show that the experimental magnetization and specific heat data
are well described if a certain type of anisotropy is included. We also present
predictions for the quasiparticle gap which might be tested in the future.
