  For a system near a quantum critical point (QCP), above its lower critical
dimension $d_L$, there is in general a critical line of second order phase
transitions that separates the broken symmetry phase at finite temperatures
from the disordered phase. The phase transitions along this line are governed
by thermal critical exponents that are different from those associated with the
quantum critical point. We point out that, if the effective dimension of the
QCP, $d_{eff}=d+z$ ($d$ is the Euclidean dimension of the system and $z$ the
dynamic quantum critical exponent) is above its upper critical dimension $d_C$,
there is an intermingle of classical (thermal) and quantum critical
fluctuations near the QCP. This is due to the breakdown of the generalized
scaling relation $\psi=\nu z$ between the shift exponent $\psi$ of the critical
line and the crossover exponent $\nu z$, for $d+z>d_C$ by a \textit{dangerous
irrelevant interaction}. This phenomenon has clear experimental consequences,
like the suppression of the amplitude of classical critical fluctuations near
the line of finite temperature phase transitions as the critical temperature is
reduced approaching the QCP.
