Renormalization of Quantum Field Theories on Noncommutative R^d, I. Scalars
Abstract
We show that a massive scalar quantum field theory on noncommutative R^d is perturbatively renormalizable if and only if the corresponding quantum field theory on commutative R^d is renormalizable. 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 satisfy the conditions of the convergence theorem. Finally, we show that the recursive procedure is equivalent to the counterterm approach.Keywords
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