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) . The Heisenberg operators can be written in terms of Schrödinger operators as and the creation operator is , where is the grand-canonical Hamiltonian. Similarly, for the imaginary-time operators, [Note that the imaginary-time creation operator is not the Hermitian conjugate of the annihilation operator .] In real time, the -point Green function is defined by where we have used a condensed notation in which signifies and signifies . The operator denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left. In imaginary time, the corresponding definition is where signifies . (The imaginary-time variables are restricted to the range from to the inverse temperature .) Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point () thermal Green function for a free particle is and the retarded Green function is where is the Matsubara frequency. Throughout, is for bosons and for fermions and denotes either a commutator or anticommutator as appropriate. (See below for details.) The Green function with a single pair of arguments () is referred to as the two-point function, or propagator.