  We develop a general formalism to calculate the force between beads attached
to a flat $d$-dimensional membrane due to the quantum fluctuations of the
membrane. The interaction potential is derived as a function of $d$ and the
membrane energy density, tension, stiffness and temperature. We find that the
induced interactions turn off when $d$ exceeds a certain critical dimension.
The potential is attractive in all cases where it is non-zero and at finite
temperature falls off exponentially for large distances.
