{"@context":{"@vocab":"https://cir.nii.ac.jp/schema/1.0/","rdfs":"http://www.w3.org/2000/01/rdf-schema#","dc":"http://purl.org/dc/elements/1.1/","dcterms":"http://purl.org/dc/terms/","foaf":"http://xmlns.com/foaf/0.1/","prism":"http://prismstandard.org/namespaces/basic/2.0/","cinii":"http://ci.nii.ac.jp/ns/1.0/","datacite":"https://schema.datacite.org/meta/kernel-4/","ndl":"http://ndl.go.jp/dcndl/terms/","jpcoar":"https://github.com/JPCOAR/schema/blob/master/2.0/"},"@id":"https://cir.nii.ac.jp/crid/1362825896074450304.json","@type":"Article","productIdentifier":[{"identifier":{"@type":"DOI","@value":"10.1063/1.532860"}},{"identifier":{"@type":"URI","@value":"https://pubs.aip.org/aip/jmp/article-pdf/40/5/2201/19231175/2201_1_online.pdf"}}],"dc:title":[{"@value":"𝓟𝓣-symmetric quantum mechanics"}],"description":[{"type":"abstract","notation":[{"@value":"<jats:p>This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal 𝒫𝒯, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1&lt;ε&lt;0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because 𝒫𝒯 symmetry is spontaneously broken. The phase transition that occurs at ε=0 manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H=p2+x2N(ix)ε with N integer and ε&gt;−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.</jats:p>"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1382825896074450305","@type":"Researcher","foaf:name":[{"@value":"Carl M. Bender"}],"jpcoar:affiliationName":[{"@value":"Department of Physics, Washington University, St. Louis, Missouri 63130"}]},{"@id":"https://cir.nii.ac.jp/crid/1382825896074450304","@type":"Researcher","foaf:name":[{"@value":"Stefan Boettcher"}],"jpcoar:affiliationName":[{"@value":"Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Maxico 87545"},{"@value":"CTSPS, Clark Atlanta University, Atlanta, Georgia 30314"}]},{"@id":"https://cir.nii.ac.jp/crid/1382825896074450176","@type":"Researcher","foaf:name":[{"@value":"Peter N. Meisinger"}],"jpcoar:affiliationName":[{"@value":"Department of Physics, Washington University, St. Louis, Missouri 63130"}]}],"publication":{"publicationIdentifier":[{"@type":"PISSN","@value":"00222488"},{"@type":"EISSN","@value":"10897658"}],"prism:publicationName":[{"@value":"Journal of Mathematical Physics"}],"dc:publisher":[{"@value":"AIP Publishing"}],"prism:publicationDate":"1999-05-01","prism:volume":"40","prism:number":"5","prism:startingPage":"2201","prism:endingPage":"2229"},"reviewed":"false","url":[{"@id":"https://pubs.aip.org/aip/jmp/article-pdf/40/5/2201/19231175/2201_1_online.pdf"}],"createdAt":"2002-07-26","modifiedAt":"2024-02-06","relatedProduct":[{"@id":"https://cir.nii.ac.jp/crid/1050292815315045376","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Exceptional band touching for strongly correlated systems in 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