Optimal alignment of Lorentz orientation and generalization to matrix Lie groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917300128448512 |
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| author | Sha, Congzhou M |
| author_facet | Sha, Congzhou M |
| contents | There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames $A$ and $B$, and given (possibly noisy) measurements of a set of 4-vectors $\{v_i\}$ made in those reference frames with components $\{v_{A,i}\}$ and $\{v_{B,i}\}$, find the optimal Lorentz transformation $Λ$ such that $Λv_{A,i}=v_{B,i}$. The first method is direct least squares optimization through a parametrization of $SO(3,1)_+$ in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal alignment of Lorentz orientation and generalization to matrix Lie groups Sha, Congzhou M Numerical Analysis High Energy Physics - Phenomenology Mathematical Physics Optimization and Control Computational Physics There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames $A$ and $B$, and given (possibly noisy) measurements of a set of 4-vectors $\{v_i\}$ made in those reference frames with components $\{v_{A,i}\}$ and $\{v_{B,i}\}$, find the optimal Lorentz transformation $Λ$ such that $Λv_{A,i}=v_{B,i}$. The first method is direct least squares optimization through a parametrization of $SO(3,1)_+$ in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups. |
| title | Optimal alignment of Lorentz orientation and generalization to matrix Lie groups |
| topic | Numerical Analysis High Energy Physics - Phenomenology Mathematical Physics Optimization and Control Computational Physics |
| url | https://arxiv.org/abs/2506.14994 |