2002/06/03 by Raimar Wulkenhaar, Wulkenhaar, Raimar
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0206018
LaTeX, 27 pages, 3 figures, uses feynmf package. v2: references and some clarifications added. v3: view of *-products changed, additional references added. To appear in proceedings of Hesselberg 2002 workshop on "Theory of renormalisation and regularisation", eds. R. Nest, F. Scheck and E. Vogt
openalex publication_date 2002/06/03 · arxiv created 2002/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I recall the main motivation to study quantum field theories on noncommutative spaces and comment on the most-studied example, the noncommutative R4. That algebra is given by the *-product which can be written in (at least) two ways: in an integral form or an exponential form. These two forms of the *-product are adapted to different classes of functions, which, when using them to formulate field theory, lead to two versions of quantum field theories on noncommutative R4. The integral form requires functions of rapid decay and a (preferably smooth) cut-off in the path integral, which therefore should be evaluated by exact renormalisation group methods. The exponential form is adapted to analytic functions with arbitrary behaviour at infinity, so that Feynman graphs can be used to compute the path integral (without cut-off) perturbatively.