Dyson's equation

The Dyson equations are relationships found by Freeman Dyson between different S-matrix elements or Green functions of a quantum field theory . Although Dyson's equations were only found for two-point and three-point functions in quantum electrodynamics by adding up an infinite number of Feynman diagrams , these integral equations are generally valid in quantum field theories and are also used for general n-point functions.

They represent the fully ( dressed ) renormalized Green functions through an interaction-free part, the so-called bare ( bare ) Green functions, and an interactive part that contains all possible interactions of the fields involved.

The Dyson equation of the electron propagator

The original Dyson equations are:

• for the electron - propagator : ${\displaystyle S=S_{0}+S_{0}\Sigma S}$
• for the photon - propagator : ${\displaystyle D^{\mu \nu }=D_{0}^{\mu \nu }+D_{0}^{\mu \alpha }\Pi _{\alpha \beta }D^{\beta \nu }}$
• for the electron - photon - vertex : ${\displaystyle \Gamma _{\mu }=\gamma _{\mu }+\Lambda _{\mu }}$

in which

• the subscript 0 denotes the interaction-free terms and
• the capital Greek letters each represent the irreducible Green function for the one-particle system, i.e.
  • ${\displaystyle \Sigma }$the electron self-energy
  • ${\displaystyle \Pi _{\alpha \beta }}$the photon vacuum polarization .

The first two equations are single-particle cases (n = 1) of the general form for n  particles , which is now often referred to as the Dyson equation :

${\displaystyle G^{n}=G_{0}^{n}+G_{0}^{n}K^{n}G^{n}}$

With

• the full green function ${\displaystyle G^{n}}$
• the Green function for n interaction- free particles ${\displaystyle G_{0}^{n}}$
• the irreducible interactions . ${\displaystyle K^{n}}$

The Dyson equation, also in the form of the Dyson-Schwinger equations , is used today in many areas of theoretical physics .

See also

• Bethe-Salpeter equation

Individual evidence

1. ^ ^{A b} F. Dyson: The S Matrix in Quantum Electrodynamics . In: Phys. Rev. . 75, 1949, p. 1736. doi : 10.1103 / PhysRev . 75.1736 .