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 original Dyson equations are:
- for the electron - propagator :
- for the photon - propagator :
- for the electron - photon - vertex :
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.
- the electron self-energy
- 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 :
With
- the full green function
- the Green function for n interaction- free particles
- the irreducible interactions .
The Dyson equation, also in the form of the Dyson-Schwinger equations , is used today in many areas of theoretical physics .
See also
Individual evidence
- ^ A b F. Dyson: The S Matrix in Quantum Electrodynamics . In: Phys. Rev. . 75, 1949, p. 1736. doi : 10.1103 / PhysRev . 75.1736 .