  Despite the fact that conserved currents have dimensions that are determined
solely by dimensional analysis (and hence no anomalous dimensions), Nature
abounds in examples of anomalous diffusion in which $L\propto t^\gamma$, with
$\gamma\ne 1/2$, and heat transport in which the thermal conductivity diverges
as $L^\alpha$. Aside from breaking of Lorentz invariance, the true common link
in such problems is an anomalous dimension for the underlying conserved
current, thereby violating the basic tenet of field theory. We show here that
the phenomenological non-local equations of motion that are used to describe
such anomalies all follow from Lorentz-violating gauge transformations arising
from N\"other's second theorem. The generalizations lead to a family of
diffusion and heat transport equations that systematize how non-local gauge
transformations generate more general forms of Fick's and Fourier's laws for
diffusive and heat transport, respectively. In particular, the associated
Goldstone modes of the form $\omega\propto k^\alpha$, $\alpha\in \mathbb R$ are
direct consequences of fractional equations of motion.
