Propagators and Path Integrals

TL;DR

This paper derives path-integral expressions for particle propagators in scalar and fermionic theories, linking quantum field theory with classical models, and explores world-line symmetries and their implications.

Contribution

It introduces a new path-integral formulation for propagators and analyzes the effects of world-line supersymmetry and reparametrization invariance in quantum field theories.

Findings

Derived path-integral expressions for propagators.

Identified topological fermionic excitations causing state doubling.

Proposed a bosonic formulation to eliminate extra states.

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.