Quantum mechanical path integrals in the first quantised approach to quantum field theory

TL;DR

This paper reviews the worldline formalism in quantum field theory, highlighting recent advancements in path integral representations, including fermions, color, higher spins, and non-commutative spaces, offering an alternative to traditional second quantisation methods.

Contribution

It introduces recent developments in the first quantised worldline formalism, expanding its applicability to various complex quantum field theory scenarios.

Findings

Path integral representation of open fermion lines

Use of auxiliary worldline fields for color

Extension to non-commutative space

Abstract

Perturbative quantum field theory usually uses second quantisation and Feynman diagrams. The worldline formalism provides an alternative approach based on first quantised particle path integrals, similar in spirit to string perturbation theory. Here we review the history, main features and present applications of the formalism. Our emphasis is on recent developments such as the path integral representation of open fermion lines, the description of colour using auxiliary worldline fields, incorporation of higher spin, and extension of the formalism to non-commutative space.