A theory of fractional electricity and magnetism is presented here which is capable of describing phenomena as disparate as the nonlocality of the Pippard kernel in superconductivity and anomalous dimensions for conserved currents in holographic dilatonic models. While it is a standard result in field theory that the scaling dimension of conserved currents and their associated gauge fields are determined strictly by dimensional analysis and hence cannot change under any amount of renormalization, it is also the case that the standard conservation laws for currents, $dJ=0$, remain unchanged in form if any differential operator that commutes with the total exterior derivative, $[d,\widehat{Y}]=0$, multiplies the current. Such an operator, effectively changing the dimension of the current, increases the allowable gauge transformations in electromagnetism and is at the heart of Nöther’s second theorem. However, this observation has not been exploited to generate new electromagnetisms. Here a consistent theory of electromagnetism is developed that exploits this hidden redundancy in which the standard gauge symmetry in electromagnetism is modified by the rotationally invariant operator, the fractional Laplacian. The resultant theories are shown to all allow for anomalous (nontraditional) scaling dimensions of the gauge field and the associated current. Using the Caffarelli-Silvestre theorem [Caffarelli, L., and L. Silvestre, 2007, Commun. Partial Differ. Equations 32, 1245.], its extension [La Nave, G., and P. Phillips, 2017, arXiv:1708.00863 [Commun. Math. Phys. (in press)] ] to $p$ forms and the membrane paradigm, either the boundary (UV) or horizon (IR) theory of holographic dilatonic models are shown to both be described by such fractional electromagnetic theories. The nonlocal Pippard kernel introduced to solve the problem of the Meissner effect in elemental superconductors can also be formulated as a special case of fractional electromagnetism. Because the holographic dilatonic models produce boundary theories that are equivalent to those arising from a bulk theory with a massive gauge field along the radial direction, the common thread linking both of these problems is the breaking of $U(1)$ symmetry down to ${\mathbb{Z}}_2$. The standard charge quantization rules fail when the gauge field acquires an anomalous dimension. The breakdown of charge quantization is discussed extensively in terms of the experimentally measurable modified Aharonov-Bohm effect in the strange metal phase of the cuprate superconductors.

中文翻译：

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座谈会：量子物质和高能物理中的分数电磁学

此处介绍了分数电和磁学的理论，该理论能够描述全息膨胀论模型中守恒电流在超导电性和反常尺度上与皮帕德核的非局部性完全不同的现象。尽管场论的标准结果是守恒电流及其相关规范场的缩放尺度是严格通过尺度分析确定的，因此在任何形式的重新归一化下都不会改变，但是电流的标准守恒律也是这种情况，$dĴ=0$，如果任何微分算子与总的外部导数进行折算，则形式上保持不变， $[d，\widehat{ÿ}]=0$，乘以电流。这种算子有效地改变了电流的大小，增加了电磁中允许的量规转换，并且是Nöther第二定理的核心。但是，这一发现尚未被利用来产生新的电磁学。在这里，人们开发出了一种一致的电磁学理论，该理论利用了这种隐藏的冗余，其中电磁学中的标准量规对称性由旋转不变算子分数拉普拉斯算子进行了修改。结果表明，所有理论都允许仪表场和相关电流的异常（非传统）缩放尺寸。使用Caffarelli-Silvestre定理[Caffarelli，L.和L. Silvestre，2007，Commun。部分差异。式 32，1245.]，其扩展名[La Nave，G.和P. Phillips，2017，arXiv：1708.00863 [Commun。数学。物理 （按中）]]转到$p$形式和膜范式，全息膨胀论模型的边界（UV）或水平（IR）理论均显示为由这种分数电磁理论来描述。为解决元素超导体中的迈斯纳效应问题而引入的非局部Pippard核也可以表述为分数电磁学的特殊情况。由于全息膨胀论模型产生的边界理论与沿径向方向具有大尺度场的整体理论所产生的边界理论相同，因此将这两个问题联系在一起的共同思路是断裂$ü（\mathrm{1个}）$ 对称性下降到 ${\mathbb{ž}}_{\mathrm{2个}}$。当量规字段获取异常尺寸时，标准电荷量化规则将失败。根据铜酸盐超导体的奇怪金属相中的实验可测量的改进的Aharonov-Bohm效应，广泛讨论了电荷量化的细目分类。