Propagators and Path Integrals

TL;DR

The paper develops path-integral representations for scalar and Dirac propagators by mapping them to classical point-particle models with worldline reparametrization invariance. It uses BRST quantization to implement gauge fixing and analyzes the role of worldline supersymmetry, showing that a manifestly supersymmetric formulation doubles the fermionic spectrum due to a hidden topological excitation. A bosonic representation of $\gamma_5$ is proposed to remove this doubling at the expense of manifest SUSY, with detailed treatment of the gauge-fixed propagator and its supersymmetric extensions. The work also discusses generalizations to background fields and interactions, and clarifies the origin and handling of auxiliary degrees of freedom in the worldline formalism.

Abstract

Path-integral expressions for one-particle propagators in scalar and fermionic field theories are derived, for arbitrary mass. This establishes a direct connection between field theory and specific classical point-particle models. The role of world-line reparametrization invariance of the classical action and the implementation of the corresponding BRST-symmetry in the quantum theory are discussed. The presence of classical world-line supersymmetry is shown to lead to an unwanted doubling of states for massive spin-1/2 particles. The origin of this phenomenon is traced to a `hidden' topological fermionic excitation. A different formulation of the pseudo-classical mechanics using a bosonic representation of $\gam_5$ is shown to remove these extra states at the expense of losing manifest supersymmetry.