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

TL;DR

This paper develops a topological framework for analyzing the convergence and renormalization of scalar quantum field theories on noncommutative spaces, proposing a recursive subtraction method and discussing implications for renormalizability.

Contribution

It introduces a topological convergence theorem for noncommutative Feynman graphs and a noncommutative subtraction scheme, advancing understanding of renormalization in noncommutative scalar field theories.

Findings

Proved convergence theorem for certain classes of diagrams.

Proposed a noncommutative analog of Bogoliubov-Parasiuk subtraction.

Indicated potential for renormalizability in supersymmetric noncommutative models.

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 $\Omega_d$ satisfy the conditions of the convergence theorem. For a generic scalar noncommutative quantum field theory on $\re^d$, the class $\Omega_d$ 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.