{"title":"朗道哈密顿量的非交换几何:度量方面","authors":"G. Nittis, M. Sandoval","doi":"10.3842/sigma.2020.146","DOIUrl":null,"url":null,"abstract":"This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects\",\"authors\":\"G. Nittis, M. Sandoval\",\"doi\":\"10.3842/sigma.2020.146\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.\",\"PeriodicalId\":8469,\"journal\":{\"name\":\"arXiv: Mathematical Physics\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-06-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3842/sigma.2020.146\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3842/sigma.2020.146","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.
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