  We calculate the optical conductivity using several models for unparticle or
scale-invariant matter. Within a Gaussian action for unparticles that is gauged
with Wilson lines, we find that the conductivity computed from the Kubo
formalism with vertex corrections yields no non-trivial deviation from the
free-theory result. This result obtains because at the Gaussian level,
unparticles are just a superposition of particle fields and hence any transport
property must be consistent with free theory. Beyond the Gaussian approach, we
adopt the continuous mass formulation of unparticles and calculate the Drude
conductivity directly. We show that unparticles in this context can be tailored
to yield an algebraic conductivity that scales as $\omega^{-2/3}$ with the
associated phase angle between the imaginary and real parts of
$\arctan\frac{\sigma_2}{\sigma_1}=60^\circ$ as is seen in the cuprates. Given
the recent results\cite{Donos2014,Rangamani2015,Langley} that gravitational
crystals lack a power-law optical conductivity, this constitutes the first
consistent account of the $\omega^{-2/3}$ conductivity and the phase angle seen
in optimally doped cuprates. Our results indicate that at each frequency in the
scaling regime, excitations on all energy scales contribute. Hence, incoherence
is at the heart of the power-law in the optical conductivity in strongly
correlated systems such as the cuprates.
