Feynman rules

Feynman rules
Name	Feynman rules
Caption	A simple Feynman diagram in quantum electrodynamics, governed by specific rules.
Field	Quantum field theory
Related	Path integral formulation, S-matrix, Perturbation theory (quantum mechanics)

Feynman rules. In quantum field theory, Feynman rules are a systematic procedure for translating the abstract mathematical formalism of a Lagrangian into a set of diagrammatic instructions for calculating scattering amplitudes. Developed primarily by Richard Feynman in the late 1940s, these rules provide a powerful pictorial and computational tool within the framework of perturbation theory. They allow physicists to compute the probability amplitudes for fundamental particle interactions, such as those observed at facilities like CERN, by associating mathematical factors with the elements of drawings now known as Feynman diagrams.

Overview and historical context

The development of Feynman rules was a cornerstone in the advancement of quantum electrodynamics and modern particle physics. Following the foundational work on quantum mechanics by figures like Paul Dirac and Werner Heisenberg, the need for a manageable computational method for interactions became acute. Richard Feynman, alongside contemporaries such as Julian Schwinger and Sin-Itiro Tomonaga, developed covariant formulations of QED, with Feynman's diagrammatic approach proving exceptionally intuitive. His techniques were later rigorously derived from more formal frameworks like the path integral formulation, a contribution heavily associated with Freeman Dyson who helped systematize and prove the equivalence of different approaches. The adoption of these rules revolutionized calculations in high-energy physics, enabling predictions tested in experiments from SLAC to the Large Hadron Collider.

Derivation from quantum field theory

Feynman rules are formally derived from the underlying quantum field theory using the machinery of perturbation theory and the S-matrix expansion. In the path integral formulation, pioneered by Richard Feynman and later extended in the work of Kenneth G. Wilson, the scattering amplitude is expressed as a functional integral over field configurations. Expanding the interaction part of the Lagrangian as a power series generates an infinite set of terms, each corresponding to a possible Feynman diagram. The application of Wick's theorem, a key result in the formalism of canonical quantization, systematically reduces the time-ordered products of field operators in the Dyson series into sums of contractions, which are identified with propagators. This algebraic process directly maps to the diagrammatic rules, with vertices from the interaction Lagrangian and internal lines representing propagators for fields like the photon or electron.

Types of Feynman rules

Different quantum field theories yield distinct sets of Feynman rules, though they share a common diagrammatic structure. The rules for quantum electrodynamics are the prototypical example, governing interactions of Dirac fields with the electromagnetic field. In contrast, the rules for quantum chromodynamics, the theory of the strong interaction, involve additional complexities like color charge and gluon self-interactions, incorporating the structure constants of the SU(3) gauge group. Theories involving the weak interaction, such as the Glashow–Weinberg–Salam model, include rules for massive gauge bosons like the W and Z bosons and Higgs boson. Furthermore, rules exist for scalar theories like phi-four theory, and for effective field theories used in contexts from chiral perturbation theory to studies of gravity in the framework of string theory.

Example: Quantum electrodynamics (QED)

In quantum electrodynamics, the Feynman rules in momentum space are particularly elegant. For an external photon line, one assigns a polarization vector. An external electron or positron line is associated with a spinor or adjoint spinor solution. An internal photon line, representing propagation, contributes a photon propagator in a gauge like the Feynman gauge. An internal fermion line contributes an electron propagator. Each vertex, representing the basic interaction where an electron emits or absorbs a photon, contributes a factor of the coupling constant—the fine-structure constant—and a gamma matrix from the Dirac algebra. Diagrams must also account for relative minus signs from Fermi–Dirac statistics, as in the famous example of Møller scattering, and include symmetry factors for identical particles.

Application in calculations

The primary application of Feynman rules is the perturbative calculation of scattering amplitudes and cross sections for processes in particle accelerators like the Large Hadron Collider. A physicist draws all topologically distinct Feynman diagrams for a given process, such as Bhabha scattering or Compton scattering, up to a desired order in the coupling constant. Using the rules, each diagram is translated into a complex mathematical expression. These contributions are then summed, often after employing techniques like renormalization to handle divergent integrals, a procedure advanced by Gerard 't Hooft and Martinus J. G. Veltman. The resulting amplitude is used to compute observable quantities, enabling precise tests of the Standard Model and searches for new physics beyond it, such as predictions leading to the discovery of the Higgs boson at CERN.

Generalizations and related concepts

The concept of Feynman rules has been generalized far beyond its original context in perturbative quantum electrodynamics. In quantum chromodynamics, the rules incorporate the non-Abelian structure of the SU(3) gauge group. The Schwinger–Dyson equations provide a non-perturbative framework related to the diagrammatic expansion. Within the path integral formulation, rules can be derived for theories on a discrete spacetime lattice, a technique crucial for non-perturbative studies in QCD. Furthermore, the diagrammatic philosophy extends to other areas of theoretical physics, including condensed matter physics (e.g., Fermi liquid theory diagrams), statistical mechanics, and even string theory, where worldsheet diagrams replace spacetime diagrams. The underlying mathematical structure is deeply connected to topics in algebraic geometry and combinatorics, as seen in the study of Feynman integrals and their periods.

Category:Quantum field theory Category:Theoretical physics Category:Richard Feynman