Abstract We discuss the regularized boundary state e−τ0HB⟩a e − τ 0 H | B a on two aspects in both 2D CFT and higher dimensional free field theory. One is its entanglement and correlation properties, which exhibit exponential decay in 2D CFT, the parameter 1/τ 0 works as a mass scale. The other concerns with its time evolution, i.e., e−itHe−τ0HB⟩a e − i t H e − τ 0 H | B a . We investigate the Kubo-Martin-Schwinger (KMS) condition on correlation function of local operators to detect the thermal properties. Interestingly we find the correlation functions in the initial state e−τ0HB⟩a e − τ 0 H | B a also partially satisfy the KMS condition. In the limit t → ∞, the correlators will exactly satisfy the KMS condition. We generally analyse quantum quench by a pure state and obtain some constraints on the possible form of 2-point correlation function in the initial state if assuming they satisfies KMS condition in the final state. As a byproduct we find in an large τ 0 limit the thermal property of 2-point function in e−τ0HB⟩a e − τ 0 H | B a also appears.
The Kubo-Martin-Schwinger boundary condition, which had first been introduced as a property characterizing the canonical ensemble average in quantum statistical mechanics, was later recoginized as a property characterizing the modular automorphisms in the so-called Tomita-Takesaki theory of von Neumann algebras. This miraculous link made it possible to apply powerful mathematical technique supplied by the Tomita-Takesaki theory to a study of some problems in mathematical formulation of statistical mechanics. More recently, the KMS condition in classical statistical mechanics has also been investigated. The aim of this exposition is to review for non-specialists those development of the “rigorous” treatment of statistical mechanics which are related to the KMS condition.