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Renormalization of Quantum Field Theories on Noncommutative R^d, I. Scalars

Iouri Chepelev·Radu Roiban
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Abstract

A noncommutative Feynman graph is a ribbon graph and can be drawn on a genus $g$ 2-surface with a boundary. We formulate a general convergence theorem for the noncommutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a noncommutative analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the subtracted graphs from a class $Ω_d$ satisfy the conditions of the convergence theorem. For a generic scalar noncommutative quantum field theory on $\re^d$, the class $Ω_d$ is smaller than the class of all diagrams in the theory. This leaves open the question of perturbative renormalizability of noncommutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a noncommutative analog of Wess-Zumino model is renormalizable.

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Iouri Chepelev, Radu Roiban. 2000-05-24. Renormalization of Quantum Field Theories on Noncommutative R^d, I. Scalars. https://doi.org/10.1088/1126-6708%2F2000%2F05%2F037

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