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 nonlocal equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from Noether's second theorem. The generalizations lead to a family of diffusion and heat transport equations that systematize how nonlocal 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.

• Received 30 June 2020
• Revised 6 December 2020
• Accepted 23 February 2021

DOI:https://doi.org/10.1103/PhysRevE.103.032115