Renormalization of quantum field theories on non-commutative R-d, 1. Scalars

A non-commutative Feynman graph is a ribbon graph and can be drawn on a gen us g 2-surface with a boundary. We formulate a general convergence theorem for the non-commutative Feynman graphs in topological terms and prove it fo r some classes of diagrams in the scalar field theories. We propose a non-c ommutative analog of Bogoliubov-Parasiuk's recursive subtraction formula an d show that the subtracted graphs from a class Omega (d) satisfy the condit ions of the convergence theorem. For a generic scalar non-commutative quant um field theory on R-d, the class Omega (d) is smaller than the class of al l diagrams in the theory. This leaves open the question of perturbative ren ormalizability of non-commutative field theories. We comment on how the sup ersymmetry can improve the situation and suggest that a non-commutative ana log of Wess-Zumino model is renormalizable.