Abstract
In various contexts in mathematical physics, such as out-of-equilibrium physics and the asymptotic information theory of many-body quantum systems, one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, which we denote by ∆, can be related by a perturbative series to another operator ∆0, whose logarithm is known. We set up a perturbation theory for the logarithm log ∆. It turns out that the terms in the series possess a remarkable algebraic structure, which enables us to write them in the form of nested commutators plus some “contact terms”.
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Acknowledgments
We thank Tom Faulkner and Daniel Harlow for discussions. SR would like to thank his undergrad mathematics instructor Prof. Subiman Kundu of IIT Delhi for his great lectures on functional analysis. This work is partially supported by the Office of High Energy Physics of U.S. Department of Energy under grant Contract Numbers DE-SC0012567 and DE-SC0019127. The work of NL is supported by a grant-in-aid from the National Science Foundation grant number PHY-1606531.
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ArXiv ePrint: 1811.05619
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Lashkari, N., Liu, H. & Rajagopal, S. Perturbation theory for the logarithm of a positive operator. J. High Energ. Phys. 2023, 97 (2023). https://doi.org/10.1007/JHEP11(2023)097
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DOI: https://doi.org/10.1007/JHEP11(2023)097