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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.00096 |
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| _version_ | 1866909234222858240 |
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| author | Alonso-Marroquin, Fernando Tang, Yaoyue Gharari, Fatemeh Najafi, M. N. |
| author_facet | Alonso-Marroquin, Fernando Tang, Yaoyue Gharari, Fatemeh Najafi, M. N. |
| contents | We propose integral representations and analytical solutions for the propagator of the $1+1$ dimensional Salpeter Hamiltonian, describing a relativistic quantum particle with no spin. We explore the exact Green function and an exact solution for a given initial condition, and also find the asymptotic solutions in some limiting cases. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent stochastic problem, namely the Bäumer equation. This equation describes \textit{relativistic} stochastic processes with time-changing anomalous diffusion. This Bäumer propagator corresponds to the Green function of a relativistic diffusion process that interpolates between Cauchy distributions for small times and Gaussian diffusion for large times, providing a framework for stochastic processes where anomalous diffusion is time-dependent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed-form solutions for the Salpeter equation Alonso-Marroquin, Fernando Tang, Yaoyue Gharari, Fatemeh Najafi, M. N. Quantum Physics High Energy Physics - Theory Mathematical Physics 81Txx (Primary) 60Gxx (Secondary) G.1.8 We propose integral representations and analytical solutions for the propagator of the $1+1$ dimensional Salpeter Hamiltonian, describing a relativistic quantum particle with no spin. We explore the exact Green function and an exact solution for a given initial condition, and also find the asymptotic solutions in some limiting cases. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent stochastic problem, namely the Bäumer equation. This equation describes \textit{relativistic} stochastic processes with time-changing anomalous diffusion. This Bäumer propagator corresponds to the Green function of a relativistic diffusion process that interpolates between Cauchy distributions for small times and Gaussian diffusion for large times, providing a framework for stochastic processes where anomalous diffusion is time-dependent. |
| title | Closed-form solutions for the Salpeter equation |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics 81Txx (Primary) 60Gxx (Secondary) G.1.8 |
| url | https://arxiv.org/abs/2407.00096 |