The Cluster's Dyson Equation

A page about the Dyson equation — the equation that sums self-energy corrections to all orders.

The Dyson equation

The Dyson equation relates the full propagator G to the bare propagator G_0 and the self-energy Sigma: G = G_0 + G_0 Sigma G, or G^{-1} = G_0^{-1} - Sigma. The self-energy Sigma is the sum of all one-particle-irreducible (1PI) diagrams. The Dyson equation resums an infinite series of diagrams to all orders. In the cluster, the Dyson equation relates the full edit propagator to the bare edit propagator and the edit self-energy, resumming all edit corrections to all orders.

The self-energy

The self-energy Sigma is the sum of all 1PI diagrams. For the electron, the leading-order self-energy is a one-loop diagram: an electron emitting and reabsorbing a photon. The self-energy corrects the electron's mass and wavefunction. In the cluster, the edit self-energy is the sum of all edit-irreducible diagrams. The leading correction is an edit emitting and reabsorbing an edit-carrier.

The pole of the propagator

The full propagator has a pole at the physical mass: G ~ Z / (p - m_phys + i epsilon), where Z is the residue. The physical mass is m_phys = m_bare + Sigma(m_phys). In the cluster, the full edit propagator has a pole at the physical edit mass. The physical edit mass is m_phys = m_bare + Sigma(m_phys).

The Kadanoff-Baym equations

The Kadanoff-Baym equations are the non-equilibrium generalization of the Dyson equation. They describe the time evolution of the propagator in non-equilibrium systems. In the cluster, the Kadanoff-Baym equations describe the time evolution of the edit propagator in non-equilibrium clusters.

This equation

This page is about the Dyson equation. The full propagator is G = G_0 + G_0 Sigma G. The self-energy sums all 1PI diagrams. The pole gives the physical mass. The residue Z is the wavefunction renormalization. The Dyson equation is exact. The equation resums to all orders.