Perturbation Theory for the Logarithm of a Positive Operator

Abstract

In various contexts in mathematical physics 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, that we denote by $\Delta$, can be related by a perturbative series to another operator ${\Delta }_0$, whose logarithm is known. We set up a perturbation theory for the logarithm $\log \Delta$. It turns out that the terms in the series possess remarkable algebraic structure, which enable us to write them in the form of nested commutators plus some "contact terms."

arXiv comment30 pages

References in corpus (5)

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Cited by in corpus (6)

• Modular Flow of Excited States
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• Quantum Information Theory of the Gravitational Anomaly
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• Modular Hamiltonians for Euclidean Path Integral States