On the Problem of Defining Charge Operators for the Dirac Quantum Field
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913428509032448 |
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| author | Rico, Pablo Costa Tumulka, Roderich |
| author_facet | Rico, Pablo Costa Tumulka, Roderich |
| contents | 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 $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Problem of Defining Charge Operators for the Dirac Quantum Field Rico, Pablo Costa Tumulka, Roderich Mathematical Physics Quantum Physics 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 $A$. |
| title | On the Problem of Defining Charge Operators for the Dirac Quantum Field |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2407.09126 |