On the Problem of Defining Charge Operators for the Dirac Quantum Field

TL;DR

This paper investigates the mathematical challenges in defining localized charge operators in the Dirac quantum field, revealing fundamental issues with their convergence and domain properties.

Contribution

It identifies key difficulties in rigorously defining charge operators for regions in space within the Dirac quantum field framework.

Findings

Series for $Q_A$ do not converge well

Domains of $Q_A$ cannot include vacuum states

Localized charge operators are problematic to define

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 $A\subseteq \mathbb{R}^3$ 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 $A$.