16 - Time-dependent quantum Green's functions

Summary

Introduction

Mathematically, given a linear differential operator L_x,

one encounters the solution to the inhomogeneous differential equation

Here ρ(x) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let

G(x, y) is the Green's function. This is a function of x with y a parameter. Take G(x, y) to satisfy the same boundary conditions as φ(x). Then

since

An example of Lx is, of course, the Schrödinger operator

We take ρ(x) = V(x)ψ(x, t),ψ(x, t) being the wave function and V(x) the potential operator.

We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al., 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000).

One- and two-time quantum Green's functions and their properties

Let us introduce the creation operator ψ^†(r, t) and annihilation operator ψ(r, t) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles:

The Hamiltonian operator for the particles is

and the number density of particles at rt is the operator

Note that r = (r₁ … r_N) for (1, 2, 3, …, N) and V(|r − r′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.