Colloquium : Fractional electromagnetism in quantum matter and high-energy physics

Abstract

We present here a theory of fractional electricity and 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. 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,Ŷ ] = 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 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 (nontraditional) scaling dimensions of the gauge field and the associated current. Using the Caffarelli/Silvestre(Caffarelli and Silvestre, 2007) theorem, its extension(Nave and Phillips, 2019) to p-forms 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 electromagnetic theories. 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. 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 Z 2 . 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.

Fractional glassy relaxation and convolution modules of distributions

“…Proof Equation ( 19) is inserted into (13a). The condition (21) guarantees that exchanging the order of summation and integration is permitted because it implies that |μζ β | < 1 if |ζ | ≥ r . Recognizing a geometric series as in ( 15) one arrives at…”

“…Applications of fractional calculus in physics [3,5,9,21] and mathematics [16,40,49] are enjoying an undiminished surge of attention in recent years. Determining extended domains and parameter ranges for fractional operators is oftentimes the crucial step in advanced applications.…”

Fractional diffusion theory of balanced heterogeneous neural networks

“…To develop an analytically tractable fractional Fokker-Planck equation with the required absorbing boundary conditions in neural networks, we construct a new link between the first passage time and the first passage leapover, previously regarded as related but independent concepts [45,46]. Our novel fractional formalisms can thus be used to analytically reveal the circuit mechanism of complex neural dynamics, extending the recent interest in using fractional approaches for understanding complex physical systems [31,59] to a new area.…”

Renormalization and conformal invariance of non-local quantum electrodynamics

“…The case of d = 3 and s = 1 has recently received special attention [18][19][20][21] partly due to its relation to the physics of graphene [4,22] and its possible connection to the infrared fixed point of three-dimensional QED [23][24][25][26][27][28]. In [29] the authors attempt to relate non-local Abelian gauge theories to strange metals. A study of the unitary and causal properties of (1.1) has been carried out in [30,31].…”