{"title":"Symbol Correspondence for Euclidean Systems","authors":"L. B. Natividad, Job A. Nable","doi":"10.7546/jgsp-62-2021-67-84","DOIUrl":null,"url":null,"abstract":"The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\\star$-product of phase space functions. In this article, the $\\star$-product of functions on the Euclidean motion group of rank three, $\\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\\star_s$ on $\\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometry and Symmetry in Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7546/jgsp-62-2021-67-84","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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Abstract
The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.
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