{"title":"Quantum geodesic flows on graphs","authors":"Edwin Beggs, Shahn Majid","doi":"10.1007/s11005-024-01860-6","DOIUrl":null,"url":null,"abstract":"<div><p>We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the <i>n</i>-leg star graph for <span>\\(n=2,3,4\\)</span> and find the same phenomenon as recently found for the <span>\\(A_n\\)</span> Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here <span>\\(\\sqrt{n}\\)</span>. We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line <span>\\(\\mathbb {Z}\\)</span> with a general edge-symmetric metric.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01860-6.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-024-01860-6","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the n-leg star graph for \(n=2,3,4\) and find the same phenomenon as recently found for the \(A_n\) Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here \(\sqrt{n}\). We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line \(\mathbb {Z}\) with a general edge-symmetric metric.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.