  We present here a theory of fractional electro-magnetism which is capable of
describing phenomenon as disparate as the non-locality of the Pippard kernel in
superconductivity and anomalous dimensions for conserved currents in
holographic dilatonic models. The starting point for our analysis is the
observation that the standard current conservation equations remain unchanged
if any differential operator that commutes with the total exterior derivative
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 N\"other's second theorem. Here we develop a consistent
theory of electromagnetism that exploits this hidden redundancy in which the
standard gauge symmetry in electromagnetism is modified by the rotationally
invariant operator, the fractional Laplacian. We show that the resultant
theories all allow for anomalous (non-traditional) scaling dimensions of the
gauge field and the associated current. Using well known extension theorems and
the membrane paradigm, we show that either the boundary (UV) or horizon (IR)
theory of holographic dilatonic models are both described by such fractional
electromagnetism. We also show that the non-local 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. We show that
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.
