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Perturbative Quantum Field Theory in the String-Inspired Formalism

Christian Schubert

TL;DR

This review articulates a string-inspired, first-quantized worldline framework for perturbative quantum field theory, deriving Bern-Kosower rules and Strassler's path-integral representations to compute one-loop and higher-loop amplitudes more efficiently than conventional Feynman methods. It develops unified master formulae for N-point photon and gluon amplitudes across scalar, spinor, and gluon loops, and demonstrates explicit QED and QCD calculations including vacuum polarization and four-point scattering, while clarifying gauge-invariance, boundary terms, and the link to Feynman-parameter techniques. The approach highlights advantages such as reduced dependence on loop momentum, streamlined helicity methods, and natural incorporation of background fields and non-Abelian color structure, with extensions toward multiloop formalisms via graph-based worldline Green's functions. The work thus provides a powerful, versatile toolkit for perturbative calculations in gauge theories and gravity, illuminating deep connections between string theory and field-theoretic perturbation theory with practical computational benefits.

Abstract

We review the status and present range of applications of the ``string-inspired'' approach to perturbative quantum field theory. This formalism offers the possibility of computing effective actions and S-matrix elements in a way which is similar in spirit to string perturbation theory, and bypasses much of the apparatus of standard second-quantized field theory. Its development was initiated by Bern and Kosower, originally with the aim of simplifying the calculation of scattering amplitudes in quantum chromodynamics and quantum gravity. We give a short account of the original derivation of the Bern-Kosower rules from string theory. Strassler's alternative approach in terms of first-quantized particle path integrals is then used to generalize the formalism to more general field theories, and, in the abelian case, also to higher loop orders. A considerable number of sample calculations are presented in detail, with an emphasis on quantum electrodynamics.

Perturbative Quantum Field Theory in the String-Inspired Formalism

TL;DR

This review articulates a string-inspired, first-quantized worldline framework for perturbative quantum field theory, deriving Bern-Kosower rules and Strassler's path-integral representations to compute one-loop and higher-loop amplitudes more efficiently than conventional Feynman methods. It develops unified master formulae for N-point photon and gluon amplitudes across scalar, spinor, and gluon loops, and demonstrates explicit QED and QCD calculations including vacuum polarization and four-point scattering, while clarifying gauge-invariance, boundary terms, and the link to Feynman-parameter techniques. The approach highlights advantages such as reduced dependence on loop momentum, streamlined helicity methods, and natural incorporation of background fields and non-Abelian color structure, with extensions toward multiloop formalisms via graph-based worldline Green's functions. The work thus provides a powerful, versatile toolkit for perturbative calculations in gauge theories and gravity, illuminating deep connections between string theory and field-theoretic perturbation theory with practical computational benefits.

Abstract

We review the status and present range of applications of the ``string-inspired'' approach to perturbative quantum field theory. This formalism offers the possibility of computing effective actions and S-matrix elements in a way which is similar in spirit to string perturbation theory, and bypasses much of the apparatus of standard second-quantized field theory. Its development was initiated by Bern and Kosower, originally with the aim of simplifying the calculation of scattering amplitudes in quantum chromodynamics and quantum gravity. We give a short account of the original derivation of the Bern-Kosower rules from string theory. Strassler's alternative approach in terms of first-quantized particle path integrals is then used to generalize the formalism to more general field theories, and, in the abelian case, also to higher loop orders. A considerable number of sample calculations are presented in detail, with an emphasis on quantum electrodynamics.

Paper Structure

This paper contains 26 sections, 49 equations, 17 figures.

Table of Contents

  1. Introduction: Strings vs. Particles, First vs. Second Quantization
  2. The Bern -- Kosower Formalism
  3. The Infinite String Tension Limit
  4. The Bern-Kosower Rules for Gluon Scattering
  5. Worldline Path Integral Representations for Effective Actions
  6. Scalar Field Theory
  7. Scalar Quantum Electrodynamics
  8. Spinor Quantum Electrodynamics
  9. Non-Abelian Gauge Theory
  10. Scalar Loop Contribution to the Gluon Effective Action
  11. Spinor Loop Contribution to the Gluon Effective Action
  12. Gluon Loop Contribution to the Gluon Effective Action
  13. Calculation of One-Loop Amplitudes
  14. The N - point Amplitude in Scalar Field Theory
  15. Photon Scattering in Quantum Electrodynamics
  16. ...and 11 more sections

Figures (17)

  • Figure 1: The loop expansion in string perturbation theory.
  • Figure 2: Vertex operators inserted on the boundary of the annulus.
  • Figure 3: Infinite string tension limit of a string diagram.
  • Figure 4: Sum of Feynman diagrams represented by a single path integral.
  • Figure 5: Expanding the path integral in powers of the background field.
  • ...and 12 more figures