Worldline Green Functions for Arbitrary Feynman Diagrams

TL;DR

The paper develops a general first-quantized worldline framework to compute scalar amplitudes on arbitrary 1D topologies by introducing a Green function $\tilde{G}(\tau,\tau')$ solved via an exact 1D electric-circuit analogy. The key result is the compact expression $\tilde{G}(\tau,\tau') = -\frac{1}{2} s + \frac{1}{2} \mathbf{v}^T \Omega^{-1} \mathbf{v}$, where $s$, $\mathbf{v}$, and the period matrix $\Omega$ depend only on topology and edge lengths, enabling construction of amplitudes $\mathscr{A}(M,N)$ together with vacuum-bubble factors $\mathscr{V}_{M}(T_a) = \exp\left[-\frac{1}{2} (\sum_a T_a) m^2\right] (\det \Omega)^{-D/2}$. Examples on lines, circles, and two-loop topologies illustrate the method, which generalizes scalar field theory amplitudes to arbitrary loop orders and reveals a Green-function structure reminiscent of the bosonic-string worldsheet. The approach complements the standard second-quantized formalism and points to extensions to spinning particles and deeper connections to string theory and Landau singularities.

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.

Figures (6)

• Figure 1: (A) One-loop diagram with $N$ external lines. The topology of a circle has one modulus, the circumference $T$ of the loop. Also there is one residual symmetry, which has to be fixed by taking off one of the proper-time integrals. (B) Two-loop diagrams with $N$ external lines. The topology of this kind has three moduli, the lengths $T_1$, $T_2$ and $T_3$ of the three edges. And there is no unfixed residual symmetry.
• Figure 2: (A) Two-loop topological space with a unit positive charge at $\tau'$ and a unit negative charge uniformly distributed over the whole space. (B) Two-loop topological space with a unit positive charge at $\tau'$ and a unit negative charge at $\tau$. (C) Two-loop topological space with a unit positive charge at $\tau$ and a unit negative charge uniformly distributed over the whole space.
• Figure 3: (A) Two-loop topological space with a unit positive charge at $\tau'$ and a unit negative charge at $\tau$. (B) Two-loop topological space with a unit negative charge at $\tau'$ and a unit positive charge at $\tau$. (C) Two-loop topological space with a half unit positive charge at $\tau'$ and a half unit negative charge at $\tau$.
• Figure 4: (A) Two-loop topological space with a half unit positive charge at $\tau'$ and a half unit negative charge at $\tau$. The lengths of the three arcs are $T_1$, $T_2$, $T_3$. $\tau'$ and $\tau$ are respectively on $T_1$ and $T_3$ and denote the lengths from the origin. The magnitudes of the electric field on each part of the space are denoted by $a$ - $e$ and the directions are chosen arbitrarily. (B) Two-loop circuit with a half unit current input at $\tau'$ and withdrawn at $\tau$. The resistances of the three arcs are $T_1$, $T_2$, $T_3$. $\tau'$ and $\tau$ are respectively on $T_1$ and $T_3$ and denote the resistance from the origin. The currents on the parts of the circuit are denoted by $a$ - $e$ and the directions are chosen arbitrarily.
• Figure 5: The topology of a line with length $T$. There is no loop and hence no period matrix nor vector $\mathbf{v}$. The only path between $\tau$ and $\tau'$ is the edge connecting the two vertices $e_1$, so we choose it as the reference path.