Quantum Euclidean spaces with noncommutative derivatives

Li Gao, Marius Junge, Edward McDonald

Research output: Contribution to journalArticlepeer-review


Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semifinite spectral triple with nonflat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a local index formula in our setting via the computation of the Connes-Chern character of the corresponding spectral triple.

Original languageEnglish (US)
Pages (from-to)153-213
Number of pages61
JournalJournal of Noncommutative Geometry
Issue number1
StatePublished - 2022


  • Moyal plane
  • Quantum Euclidean space
  • local index formula
  • pseudo-differential operators

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Mathematical Physics
  • Geometry and Topology


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