{"title":"𝒫𝒯-symmetric quantum mechanics","authors":"Carl M. Bender, Daniel W. Hook","doi":"10.1103/revmodphys.96.045002","DOIUrl":null,"url":null,"abstract":"It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a <mjx-container ctxtmenu_counter=\"246\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container>-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of <mjx-container ctxtmenu_counter=\"247\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container>-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of <mjx-container ctxtmenu_counter=\"248\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container> symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of <mjx-container ctxtmenu_counter=\"249\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container> symmetry by focusing on elementary one-dimensional <mjx-container ctxtmenu_counter=\"250\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container>-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of <mjx-container ctxtmenu_counter=\"251\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-font=\"script\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"script upper P upper T\" data-semantic-type=\"identifier\"><mjx-c noic=\"true\" style=\"padding-top: 0.703em;\">𝒫</mjx-c><mjx-c style=\"padding-top: 0.703em;\">𝒯</mjx-c></mjx-mi></mjx-math></mjx-container>-symmetric quantum theory.","PeriodicalId":21172,"journal":{"name":"Reviews of Modern Physics","volume":null,"pages":null},"PeriodicalIF":45.9000,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reviews of Modern Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/revmodphys.96.045002","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a 𝒫𝒯-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of 𝒫𝒯-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of 𝒫𝒯 symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of 𝒫𝒯 symmetry by focusing on elementary one-dimensional 𝒫𝒯-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of 𝒫𝒯-symmetric quantum theory.
期刊介绍:
Reviews of Modern Physics (RMP) stands as the world's foremost physics review journal and is the most extensively cited publication within the Physical Review collection. Authored by leading international researchers, RMP's comprehensive essays offer exceptional coverage of a topic, providing context and background for contemporary research trends. Since 1929, RMP has served as an unparalleled platform for authoritative review papers across all physics domains. The journal publishes two types of essays: Reviews and Colloquia. Review articles deliver the present state of a given topic, including historical context, a critical synthesis of research progress, and a summary of potential future developments.