Luttinger–Ward functional

In solid state physics, the Luttinger–Ward functional, proposed by Joaquin Mazdak Luttinger and John Clive Ward in 1960, is a scalar functional of the bare electron-electron interaction and the renormalized one-particle propagator. In terms of Feynman diagrams, the Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible diagrams, i.e., all diagrams without particles going in or out that do not fall apart if one removes two propagator lines. It is usually written as ${\displaystyle \Phi [G]}$ or ${\displaystyle \Phi [G,U]}$, where ${\displaystyle G}$ is the one-particle Green's function and ${\displaystyle U}$ is the bare interaction.

The Luttinger–Ward functional has no direct physical meaning, but it is useful in proving conservation laws.

The functional is closely related to the Baym–Kadanoff functional constructed independently by Gordon Baym and Leo Kadanoff in 1961. Some authors use the terms interchangeably; if a distinction is made, then the Baym–Kadanoff functional is identical to the two-particle irreducible effective action ${\displaystyle \Gamma [G]}$, which differs from the Luttinger–Ward functional by a trivial term.