{"title":"离散圆柱的积分量化","authors":"J. Gazeau, R. Murenzi","doi":"10.3390/quantum4040026","DOIUrl":null,"url":null,"abstract":"Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.","PeriodicalId":34124,"journal":{"name":"Quantum Reports","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Integral Quantization for the Discrete Cylinder\",\"authors\":\"J. Gazeau, R. Murenzi\",\"doi\":\"10.3390/quantum4040026\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.\",\"PeriodicalId\":34124,\"journal\":{\"name\":\"Quantum Reports\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-08-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quantum Reports\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/quantum4040026\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Physics and Astronomy\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Reports","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/quantum4040026","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 4
摘要
协变积分量化是基于正则化正算子值测度(POVM)的连续或离散族对恒等的解析,POVM具有吸引人的概率内容,并以协变方式变换。它们的优点之一是能够避免由于经典模型中存在奇点而引起的问题。本文对相空间为Z×S1的系统,即在圆上运动的系统,实现了协变积分量子化。该相空间的对称群是Weyl-Heisenberg群的离散紧化版本,即阿贝尔群的中心扩展Z×SO(2)。在这方面,相空间被视为具有其中心的群的右傍集。这个群的非平凡酉不可约表示作用于L2(S1),在相空间上是平方可积的。我们展示了如何从相空间上的(权)函数推导出相应的协变积分量子化,并由此得到恒等的解析。作为后者的特殊情况,我们用de bi -del Olmo-Gonzales和Kowalski-Rembielevski-Papaloucas在圆上的相干态恢复量子化。我们的方法的另一个直接结果是穆昆达维格纳变换。我们还研究了由移位高斯、冯·米塞斯、泊松和费杰迈尔核构建的相干态的具体情况。恒星表示的应用正在进行中。
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.