2007/08/02 by T. Kopf, M. Paschke, Mario Paschke
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Covariant transformation #Feynman diagram #Mathematical physics #Mathematics #Noncommutative algebraic geometry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Physics #Pure mathematics #Quantum differential calculus #Quantum mechanics #math-ph #math.MP
paper · pdf · doi:10.1063/1.2804075
20 pages
arxiv created 2007/08/02 · openalex publication_date 2007/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We generalize the previously given algebraic version of “Feynman’s proof of Maxwell’s equations” to noncommutative configuration spaces. By doing so, we also obtain an axiomatic formulation of nonrelativistic quantum mechanics over such spaces, which, in contrast to most examples discussed in the literature, does not rely on a distinguished set of coordinates. We give a detailed account of several examples, e.g., C∞(Q)⊗Mn(C) which leads to non-Abelian Yang-Mills theories, and of noncommutative tori Tθd. Moreover, we examine models over the Moyal-deformed plane Rθ2. Assuming the conservation of electrical charges, we show that in this case the canonical uncertainty relation [xk,ẋl]=igkl with metric gkl is only consistent if gkl is constant.