Non-Hermitian spacetime and generalized thermofield double formalism

TL;DR

This paper investigates non-Hermitian transition matrices in quantum field theories and gravity, establishing their role in the thermofield double formalism, black hole entropy, and complex metrics within the AdS/CFT framework.

Contribution

It introduces the concept of $ ext{eta}$-pseudo-Hermitian transition matrices and explores their significance in holography and black hole physics, extending the thermofield double formalism.

Findings

Non-Hermitian transition matrices can be interpreted as density matrices.

Pseudoentropy of transition matrices relates to black hole entropy.

Strong subadditivity of pseudoentropy is established.

Abstract

In this paper, we explore the non-Hermitian transition matrix and its gravity dual. States in quantum field theories or gravity theories are typically prepared using Euclidean path integrals. We demonstrate that it is both natural and necessary to introduce non-Hermitian transitions to describe the state when employing different inner products in Euclidean quantum field theories. Transition matrices that are $\eta$-pseudo-Hermitian, with $\eta$ being positive-definite, play the same role as density matrices, where the operator $\eta$ is closely related to the definition of the inner product. Moreover, there exists a one-to-one correspondence between these transition matrices and density matrices. In the context of AdS/CFT correspondence, the Euclidean path integral in the boundary field theory can be translated to the bulk gravitational path integral. We provide an overview of the construction and interpretation of non-Hermitian spacetime. Specifically, we demonstrate the crucial role of the non-Hermitian transition matrix in realizing the thermofield concept in general cases and in understanding the gravity states dual to the eternal black hole. In this context, the pseudoentropy of the transition matrix can also be interpreted as black hole entropy. Finally, we highlight the strong subadditivity property of pseudoentropy, and the connection between non-Hermitian transition matrices and complex metrics.