From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields

TL;DR

This paper introduces a new approach to constructing star products for scalar fields using Kontsevich graphs, establishing a correspondence with Feynman graphs and extending concepts in quantum field theory.

Contribution

It develops a novel method for star products at multiple levels, linking Kontsevich graphs to Feynman graphs, and generalizes existing star product frameworks in quantum field theory.

Findings

Established a one-to-one correspondence between Kontsevich graphs and Feynman graphs.

Constructed star products at three levels: functions, fields, and functionals.

Extended the framework of star products in perturbative algebraic quantum field theory.

Abstract

In the present paper we construct the star products concerning scalar fields in the covariant case from a new approach. We construct the star products at three levels, which are levels of functions on Rd, fields and functionals respectively. We emphases that the star product at level of functions is essence and starting point for our setting. Firstly the star product of functions includes all algebraic and combinatorial information of the star products concerning the scalar fields and functionals almost. Secondly, a more interesting point is that the star product of functions concerns only finite dimensional issue, which is a Moyal-like star product on Rd generated by a bi-vector field with abstract coefficients. Thus the Kontsevich graphs play some roles naturally. Actually we prove that there is an ono-one correspondence between a class of Kontsevich graphs and the Feynman graphs. Additionally the Wick theorem, Wick power and the expectation of Wick-monomial are discussed in terms of the star product at level of functions. Our construction can be considered as the generalisation of the star products in perturbative algebraic quantum fields theory and twist product introduced in [1],[2].