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

TL;DR

This paper surveys the worldline formalism, a first-quantised path-integral approach to quantum field theory, highlighting its advantages over standard second-quantised perturbation theory and its connections to string theory. It presents three calculation strategies—worldline numerics, semiclassical worldline instantons, and analytic Gaussian (string-inspired) evaluation—and shows how to compute scalar and spinor QED, non-abelian gauge theory, gravity, higher spin, and non-commutative spacetime within a unified framework. Key contributions include master formulae for one-loop N-photon and N-gluon amplitudes, cycle replacement rules for spin, auxiliary colour fields for non-abelian colour encodings, and extensions to constant external fields, multi-loop sewing, and curved space; these yield computationally efficient, gauge-invariant representations. The work demonstrates the practical reach of worldline methods through applications to Schwinger pair production, Euler–Heisenberg actions, and gravity-coupled amplitudes, with implications for high-precision QED, non-perturbative phenomena, and non-commutative field theories.

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.

Figures (3)

• Figure 1: Multi-photon Compton scattering diagram.
• Figure 2: Diagrams contributing to the 2-loop QED $\beta$ function.
• Figure 3: Ladder and crossed-ladder diagrams in scalar field theory.