Green's function (many-body theory)

In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point 'Green's functions' in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)

We consider a many-body theory with field operator (annihilation operator written in the position basis) ${\displaystyle \psi (\mathbf {x} )}$.

The Heisenberg operators can be written in terms of Schrödinger operators as

and ${\displaystyle {\bar {\psi }}(\mathbf {x} ,t)=[\psi (\mathbf {x} ,t)]^{\dagger }}$, where ${\displaystyle K=H-\mu N}$ is the grand-canonical Hamiltonian.

...
Wikipedia