Spin in the Worldline Path Integral in 2+1 Dimensions

TL;DR

This paper develops a worldline path integral formulation for Dirac fields in 2+1 dimensions, explicitly incorporating spin effects through a novel spin action component.

Contribution

It introduces a new worldline action for 2+1D Dirac fields that explicitly includes spin, derived by integrating out an auxiliary gauge variable.

Findings

Derived a worldline action combining kinetic and spin terms.

Confirmed the spin action matches expected spin-1/2 behavior.

Provided a framework for analyzing Dirac fields in lower dimensions.

Abstract

We construct a worldline path integral for the effective action and propagator of a Dirac field in 2+1 dimensions in an Abelian gauge field background. Integrating over an auxiliary gauge group variable we derive a worldline action depending only on $x(\tau)$, the spacetime paths. We show that that action is a combination of a kinetic term plus a spin action. The first is proportional to $\delta[\dot{x}^2(\tau)- 1]$. The second agrees exactly with the spin action one should expect for a spin-1/2 field.