[Paper Review] An Introduction to the Worldline Technique for Quantum Field Theory Calculations

Christian Schubert|arXiv (Cornell University)|Oct 15, 1996

Quantum Electrodynamics and Casimir Effect8 citations

TL;DR

This paper introduces the worldline technique—a string-inspired method for perturbative quantum field theory calculations—demonstrating its application to photon scattering in QED up to three loops. It provides a pedagogical overview of the formalism, emphasizing its efficiency and connection to string theory, with detailed computations showing its viability for higher-order amplitudes in QED.

ABSTRACT

These two lectures give a pedagogical introduction to the ``string-inspired'' worldline technique for perturbative calculations in quantum field theory. This includes an overview over the present range of its applications. Several examples are calculated in detail, up to the three-loop level. The emphasis is on photon scattering in quantum electrodynamics.

Motivation & Objective

• To provide a pedagogical introduction to the worldline technique for researchers in quantum field theory.
• To demonstrate the method's applicability to higher-order perturbative calculations, particularly in quantum electrodynamics.
• To bridge insights from string theory with practical quantum field theory computations.
• To illustrate the technique through explicit, step-by-step calculations up to the three-loop level.
• To establish the worldline method as a viable alternative to traditional Feynman diagram techniques for certain amplitudes.

Proposed method

• The worldline technique reformulates quantum field theory amplitudes using path integrals over particle worldlines, inspired by string theory formalism.
• It employs a proper-time representation of the propagator, replacing the standard momentum-space Feynman integrals with worldline path integrals.
• The method uses the worldline action to compute effective actions and amplitudes via functional determinants and trace operations.
• Key computations involve the evaluation of worldline path integrals for scalar and spinor fields, with gauge invariance preserved through appropriate vertex operators.
• The formalism allows for the systematic computation of scattering amplitudes by integrating over worldline configurations with appropriate interaction vertices.
• The technique is applied to photon-photon scattering in QED, with results derived up to three-loop order using explicit path integral evaluations.

Experimental results

Research questions

• RQ1 How can string-inspired worldline methods be systematically applied to quantum field theory calculations in QED?
• RQ2 What is the structure of the worldline path integral formulation for higher-order amplitudes in QED?
• RQ3 How does the worldline technique compare in efficiency and clarity to standard Feynman diagram methods for photon scattering?
• RQ4 What are the technical challenges and solutions in extending the worldline method to three-loop order?
• RQ5 In what ways does the worldline formalism preserve gauge invariance and unitarity in perturbative calculations?

Key findings

• The worldline technique successfully computes photon-photon scattering amplitudes in QED up to three-loop order using path integral methods.
• The method provides a compact and efficient alternative to traditional Feynman diagram techniques for certain amplitudes.
• The formalism naturally incorporates gauge invariance through the use of proper-time regularization and worldline vertex operators.
• Explicit calculations show that the worldline approach yields results consistent with standard field theory at the three-loop level.
• The technique demonstrates strong connections to string theory, particularly in the structure of the worldline action and path integral measure.
• The paper establishes the worldline method as a viable and insightful tool for higher-loop calculations in quantum field theory.