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

2024/07/12 by Pablo Costa Rico, Roderich Tumulka, Rico, Pablo Costa +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Crystallography and Radiation Phenomena #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2407.09126

openalex publication_date 2024/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• What is electric charge?: Charge as a local observable in relativistic quantum field theory

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