N\"other's Second Theorem as an Obstruction to Charge Quantization

Abstract

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, $[d,\hat Y]=0$, 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 N\"other's second theorem. We review here our recent work on one particular instance of this theorem, namely fractional electromagnetism and highlight the holographic dilaton models that exhibit such behavior and the physical consequences this theory has for charge quantization. Namely, the standard electromagnetic gauge and the fractional counterpart cannot both yield integer values of Planck's constant when they are integrated around a closed loop, thereby leading to a breakdown of charge quantization.