Perturbation theory for the logarithm of a positive operator
Nima Lashkari (School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA); Hong Liu (Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA); Srivatsan Rajagopal (Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA)
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”.