fermionpropagator

A fermion propagator, sometimes called a fermionpropagator, is a Green's function in quantum field theory that describes the amplitude for a spin-1/2 fermion to propagate from one spacetime point to another. It encodes the spinor structure and the causal propagation of fermionic degrees of freedom and is the inverse of the Dirac operator.

In Minkowski space, for a free Dirac fermion of mass m, the Feynman propagator S_F(x − y) satisfies (i γ^μ ∂_μ − m) S_F(x − y) = δ^4(x − y). In momentum space, it takes the form S_F(p) = i (p_μ γ^μ + m) / (p^2 − m^2 + iε). For a massless fermion, S_F(p) = i p_μ γ^μ / (p^2 + iε). The propagator can be represented as a Fourier transform between momentum and position spaces.

Propagators are central to perturbation theory, where fermion lines connect interaction vertices in Feynman diagrams and propagate quantum information between events. In the presence of gauge fields, the Dirac operator generalizes to i D/ − m with D_μ = ∂_μ − i g A_μ^a T^a, so the propagator becomes the inverse of i D/ − m, denoted S_F(x, y; A). Variants include retarded and advanced propagators, which encode causal propagation, as well as Euclidean or curved-space generalizations used in different formulations.

Key properties include their role as Green’s functions for the Dirac equation, their spinor structure, and their behavior under renormalization in interacting theories.