### Abstract

A non-commutative 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 non-commutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a non-commutative 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 non-commutative quantum field theory on ℝ^{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 non-commutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a non-commutative analog of Wess-Zumino model is renormalizable.

Original language | English (US) |
---|---|

Pages (from-to) | 23-30 |

Number of pages | 8 |

Journal | Journal of High Energy Physics |

Volume | 4 |

Issue number | 5 PART B |

State | Published - Dec 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Cite this

^{d}, 1. Scalars.

*Journal of High Energy Physics*,

*4*(5 PART B), 23-30.

}

^{d}, 1. Scalars',

*Journal of High Energy Physics*, vol. 4, no. 5 PART B, pp. 23-30.

**Renormalization of quantum field theories on non-commutative ℝ ^{d}, 1. Scalars.** / Chepelev, Iouri; Roiban, Radu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Renormalization of quantum field theories on non-commutative ℝd, 1. Scalars

AU - Chepelev, Iouri

AU - Roiban, Radu

PY - 2000/12/1

Y1 - 2000/12/1

N2 - A non-commutative 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 non-commutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a non-commutative analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the subtracted graphs from a class Ωd satisfy the conditions of the convergence theorem. For a generic scalar non-commutative quantum field theory on ℝd, the class Ωd is smaller than the class of all diagrams in the theory. This leaves open the question of perturbative renormalizability of non-commutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a non-commutative analog of Wess-Zumino model is renormalizable.

AB - A non-commutative 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 non-commutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a non-commutative analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the subtracted graphs from a class Ωd satisfy the conditions of the convergence theorem. For a generic scalar non-commutative quantum field theory on ℝd, the class Ωd is smaller than the class of all diagrams in the theory. This leaves open the question of perturbative renormalizability of non-commutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a non-commutative analog of Wess-Zumino model is renormalizable.

UR - http://www.scopus.com/inward/record.url?scp=7244248830&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=7244248830&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:7244248830

VL - 4

SP - 23

EP - 30

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 5 PART B

ER -

^{d}, 1. Scalars. Journal of High Energy Physics. 2000 Dec 1;4(5 PART B):23-30.