Worldline Green Functions for Arbitrary Feynman Diagrams

TL;DR

This paper introduces a universal method to compute scalar worldline Green functions on any 1D topology, enabling the evaluation of complex 1-loop Feynman diagrams using a first-quantized approach.

Contribution

It presents a novel general technique for deriving worldline Green functions on arbitrary 1D spaces, extending the first-quantized method for Feynman diagram calculations.

Findings

Derived a compact expression for the worldline Green function.

Established an analogy with electric circuits for the Green function problem.

Enabled calculation of arbitrary 1-loop Feynman diagrams using the new method.

Abstract

We propose a general method to obtain the scalar worldline Green function on an arbitrary 1D topological space, with which the first-quantized method of evaluating 1-loop Feynman diagrams can be generalized to calculate arbitrary ones. The electric analog of the worldline Green function problem is found and a compact expression for the worldline Green function is given, which has similar structure to the 2D bosonic Green function of the closed bosonic string.