On the Renormalization of Non-Commutative Field Theories

TL;DR

This paper explores the quantization, renormalization, and algebraic properties of non-commutative field theories, demonstrating the validity of the Quantum Action Principle, extending BPHZ renormalization, and analyzing model renormalizability.

Contribution

It introduces the application of the Quantum Action Principle and BPHZ renormalization scheme to non-commutative field theories, and assesses model renormalizability using algebraic methods.

Findings

Quantum Action Principle holds for non-commutative theories

BPHZ renormalization scheme can be generalized to non-commutative cases

Certain models of self-interacting scalar fields are renormalizable

Abstract

This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense. Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.