  Magneto-quantum oscillation experiments in high temperature superconductors
show a strong thermally-induced suppression of the oscillation amplitude
approaching critical dopings---in support of a quantum critical origin of their
phase diagrams. We suggest that, in addition to a thermodynamic mass
enhancement, these experiments may directly indicate the increasing role of
quantum fluctuations that suppress the oscillation amplitude through inelastic
scattering. We show that the traditional theoretical approaches beyond
Lifshitz-Kosevich to calculate the oscillation amplitude in correlated metals
result in a contradiction with the third law of thermodynamics and suggest a
way to rectify this problem.
