A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, and acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena and subjects in quantum regimes, such as quantum resonances, superradiance, the continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, first given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

本文对非赫米特经典物理学和量子物理学的基础和应用进行了综述。首先，以教育学和数学上连贯的方式描绘了非埃尔米特线性代数中的关键定理和中心概念，包括约旦范式，双正交性，例外点，伪埃尔米特性和奇偶时间对称性。在此基础上，我们概述了如何使用各种经典系统（从光子学，力学，电路和声学到活性物质）来模拟非赫米特波物理学。特别是，我们讨论了其中发现的丰富而独特的现象，例如单向不可见性，增强的灵敏度，拓扑能量转移，相干完美吸收，单模激光和稳健的生物传输。然后，我们在Feshbach投影方法和量子轨迹方法的基础上，详细解释非Hermitian算符如何作为开放量子系统的有效描述而出现。我们将讨论它们在与各个领域相关的物理系统中的应用，包括原子，分子和光学物理学，介观物理学和核物理学，重点关注量子现象中的突出现象和主题，例如量子共振，超辐射，连续量子芝诺。效应，量子临界现象，量子色动力学中的狄拉克光谱以及非unit形共形场论。最后，我们在非Hermitian系统的复杂频谱中引入了能带拓扑的概念，并通过提供证明（首先由本综述首先以完整的方式给出）来介绍它们的分类，以及许多说明性的例子。还讨论了与非赫米特物理学有关的其他主题，包括不可逆的传输，速度限制，非单位的量子游走。