A complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions. In the onephase region, the fluctuations are normal and its algebra is non-Abelian. In the two-phase region and on the critical line (T_c>0) the momentum fluctuation is normal, the displacement is critical, and the algebra is Abelian; at T_c=0 (quantum phase transition) this algebra is non-Abelian with abnormal displacement and supernormal (squeezed) momentum fluctuation operators, both being dimension dependent.