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

• URL: http://arxiv.org/abs/2407.09126v1
• Date: Fri, 12 Jul 2024 09:44:31 GMT
• Title: On the Problem of Defining Charge Operators for the Dirac Quantum Field
• Authors: Pablo Costa Rico, Roderich Tumulka,
• Abstract summary: We ask about operators $Q_A$ representing the charge content of a region 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $A$.

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