On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917834086416384 |
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| author | Frolov, Lawrence Leigh, Samuel E. Tahvildar-Zadeh, A. Shadi |
| author_facet | Frolov, Lawrence Leigh, Samuel E. Tahvildar-Zadeh, A. Shadi |
| contents | A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac's formalism of multi-time wave functions, i.e, wave functions $Ψ(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2})$ where $\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2}$ are generic spacetime events of the photon and two electrons respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_1}\}$ and $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_2}\}$ compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06019 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions Frolov, Lawrence Leigh, Samuel E. Tahvildar-Zadeh, A. Shadi Mathematical Physics Analysis of PDEs Quantum Physics 35Q41, 81Q05, 81V10 A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac's formalism of multi-time wave functions, i.e, wave functions $Ψ(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2})$ where $\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2}$ are generic spacetime events of the photon and two electrons respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_1}\}$ and $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_2}\}$ compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times. |
| title | On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions |
| topic | Mathematical Physics Analysis of PDEs Quantum Physics 35Q41, 81Q05, 81V10 |
| url | https://arxiv.org/abs/2312.06019 |