Conjugate linear symmetry and its application to <inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M2.png"/></alternatives></inline-formula>-symmetry quantum theory
Authors: Huang Yong-Feng , Cao Huai-Xin , Wang Wen-Hua ,   School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China,   School of Mathematics, Changji University, Changji 831100, China,   School of Ethnic Education, Shaanxi Normal University, Xi’an 710119, China
Published: December 2020

The Hamiltonians of classical quantum systems are Hermitian (self-adjoint) operators. The self-adjointness of a Hamiltonian not only ensures that the system follows unitary evolution and preserves probability conservation, but also guarantee that the Hamiltonian has real energy eigenvalues. We call such systems Hermitian quantum systems. However, there exist indeed some physical systems whose Hamiltonians are not Hermitian, for instance, <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M10.png"/></alternatives></inline-formula>-symmetry quantum systems. We refer to such systems as non-Hermitian quantum systems. To discuss in depth <inline-formula><tex-math id="M11">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M11.png"/></alternatives></inline-formula>-symmetry quantum systems, some properties of conjugate linear operators are discussed first in this paper due to the conjugate linearity of the operator <inline-formula><tex-math id="M12">\begin{document}$ {\mathcal{P}}{\mathcal{T}}, $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M12.png"/></alternatives></inline-formula> including their matrix represenations, spectral structures, etc. Second, the conjugate linear symmetry and unbroken conjugate linear symmetry are introduced for linear operators, and some equivalent characterizations of unbroken conjugate linear symmetry are obtained in terms of the matrix representations of the operators. As applications, <inline-formula><tex-math id="M13">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M13.png"/></alternatives></inline-formula>-symmetry and unbroken <inline-formula><tex-math id="M14">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M14.png"/></alternatives></inline-formula>-symmetry of Hamiltonians are discussed, showing that unbroken <inline-formula><tex-math id="M15">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M15.png"/></alternatives></inline-formula>-symmetry is not closed under taking tensor-product operation by some specific examples. Moreover, it is also illustrated that the unbroken <inline-formula><tex-math id="M16">\begin{document}$ {\mathcal{P}}{\mathcal{T}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191173_M16.png"/></alternatives></inline-formula>-symmetry is neither a sufficient condition nor a necessary condition for Hamiltonian to be Hermitian under a new positive definite inner product.