论文标题
异常扩散和Noether的第二个定理
Anomalous diffusion and Noether's second theorem
论文作者
论文摘要
尽管保守的电流的尺寸仅由维数分析（因此没有异常的维度）确定，但自然在异常扩散的示例中，$ l \ propto t^γ$，$γ\ ne 1/2 $，以及热电导率差的热传输，如$ l^$ l^α$。除了破坏Lorentz的不变性外，此类问题中的真正常见联系是基础保守电流的一个异常维度，从而违反了现场理论的基本宗旨。我们在这里表明，用于描述这种异常的现象学非本地运动方程都是由洛伦兹竞争量规变换源于Nöther的第二个定理。概括导致了一个扩散和热传输方程的家族，该家族将非本地量规变换分别产生更一般形式的Fick's和Fourier定律，以进行扩散和热传输。特别是，$ω\ propto k^α$，$α\ in \ mathbb r $的相关金石模式是运动分数方程的直接后果。
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 $L\propto t^γ$, with $γ\ne 1/2$, and heat transport in which the thermal conductivity diverges as $L^α$. 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 Nöther'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 $ω\propto k^α$, $α\in \mathbb R$ are direct consequences of fractional equations of motion.
