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Parametric Representation of Noncommutative Field Theory I Introduction

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormaliz-able φ 4 4 quantum field theory on the Moyal non commutative R 4 space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual " Kirchoff " or " Symanzik " poly-nomials of commutative field theory, but contain richer topological information. Non-commutative field theories (for a general review see [1]) deserve a thorough and systematic investigation. Indeed they may be relevant for physics beyond the standard model. They are certainly effective models for certain limits of string theory [2]-[3]. What is often less emphasized is that they can also describe effective physics in our ordinary standard world but with non-local interactions, such as the physics of the quantum Hall effect [4]. In joint work with J. Magnen and F. Vignes-Tourneret [5], we provided recently a new proof that the Grosse-Wulkenhaar scalar Φ 4 theory on the Moyal space R 4 , hereafter called NCΦ 4 4 , is renormalizable to all orders in perturbation theory using direct space multiscale analysis.