Abstract
It is well known how to define the operator Q for the total charge (i.e., positron number minus electron number) on the standard Hilbert space of the second-quantized Dirac equation. Here we ask about operators Q_A representing the charge content of a region A⊆ R³ in 3d physical space. There is a natural formula for Q_A but, as we explain, there are difficulties about turning it into a mathematically precise definition. First, Q_A can be written as a series but its convergence seems hopeless. Second, we show for some choices of A that if Q_A could be defined then its domain could not contain either the vacuum vector or any vector obtained from the vacuum by applying a polynomial in creation and annihilation operators. Both observations speak against the existence of Q_A for generic A.