  Motivated by the normal state of the cuprates in which the f-sum rule
increases faster than a linear function of the particle density, we derive a
conductivity sum rule for a system in which the kinetic energy operator in the
Hamiltonian is a general function of the momentum squared. Such a kinetic
energy arises in scale invariant theories and can be derived within the context
of holography. Our derivation of the f-sum rule is based on the gauge couplings
of a non-local Lagrangian in which the kinetic operator is a fractional
Laplacian of order $\alpha$. We find that the f-sum rule in this case deviates
from the standard linear dependence on the particle density. We find two
regimes. At high temperatures and low densities, the sum rule is proportional
to $nT^{\frac{\alpha-1}{\alpha}}$ where $T$ is the temperature. At low
temperatures and high densities, the sum rule is proportional to
$n^{1+\frac{2(\alpha-1)}{d}}$ with $d$ being the number of spatial dimensions.
The result in the low temperature and high density limit, when $\alpha < 1$,
can be used to qualitatively explain the behavior of the effective number of
charge carriers in the cuprates at various doping concentrations.
