Feynman loops and three-dimensional quantum gravity

TL;DR

This paper extends Feynman diagrams to include quantum gravity effects in three dimensions, analyzing how large momenta in loops influence gravitational responses, with detailed calculations for trefoil knots revealing threshold effects and connections to knot invariants.

Contribution

It introduces a geometric approach to quantum gravity interactions in Feynman loops and explores their impact on knot invariants like the coloured Jones polynomial.

Findings

Gravitational response becomes non-trivial at high momenta thresholds.

Large momentum loops can be described by simple geometric models.

New insights into the coloured Jones polynomial limit.

Abstract

This paper explores the idea that within the framework of three-dimensional quantum gravity one can extend the notion of Feynman diagram to include the coupling of the particles in the diagram with quantum gravity. The paper concentrates on the non-trivial part of the gravitational response, which is to the large momenta propagating around a closed loop. By taking a limiting case one can give a simple geometric description of this gravitational response. This is calculated in detail for the example of a closed Feynman loop in the form of a trefoil knot. The results show that when the magnitude of the momentum passes a certain threshold value, non-trivial gravitational configurations of the knot play an important role. The calculations also provide some new information about a limit of the coloured Jones polynomial which may be of independent mathematical interest.