Zassenhaus decomposition of half-sided translations and generalizations in 2d conformal field theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917998091042816 |
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| author | Ramchander, Manish |
| author_facet | Ramchander, Manish |
| contents | We study the half-sided translations associated to Rindler wedge algebras for conformal field theories in 1+1 Minkowski spacetime, generated by an unbounded operator $\mathcal{G}$, in terms of bilinear forms $G, G'$ made from entanglement Hamiltonians of the underlying algebras such that $\mathcal{G} = G+G'$. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, $G, G'$ can be regularized using smooth bump functions to operators $\hat{G}, \hat{G}'$ with well-defined commutators, and use them to do a centered Zassenhaus expansion of $\exp(i \mathcal{G} s)$ in terms of $\hat{G}$ and $\hat{G}'$ which is tractable and respects causality. We show that in fact half-sided translations is a special case in a large class of operators $\mathcal{O}$ for which a similar decomposition can be done by defining $\mathcal{O} = O_L+O_R$ with $O_{L}, O_{R}$ chosen approriately. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zassenhaus decomposition of half-sided translations and generalizations in 2d conformal field theory Ramchander, Manish High Energy Physics - Theory Mathematical Physics We study the half-sided translations associated to Rindler wedge algebras for conformal field theories in 1+1 Minkowski spacetime, generated by an unbounded operator $\mathcal{G}$, in terms of bilinear forms $G, G'$ made from entanglement Hamiltonians of the underlying algebras such that $\mathcal{G} = G+G'$. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, $G, G'$ can be regularized using smooth bump functions to operators $\hat{G}, \hat{G}'$ with well-defined commutators, and use them to do a centered Zassenhaus expansion of $\exp(i \mathcal{G} s)$ in terms of $\hat{G}$ and $\hat{G}'$ which is tractable and respects causality. We show that in fact half-sided translations is a special case in a large class of operators $\mathcal{O}$ for which a similar decomposition can be done by defining $\mathcal{O} = O_L+O_R$ with $O_{L}, O_{R}$ chosen approriately. |
| title | Zassenhaus decomposition of half-sided translations and generalizations in 2d conformal field theory |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2408.00164 |