Green’s Functions

Abstract

Consider a Hamiltonian of the system of particles interacting via a pair potential Φ and situated in the entire three-dimensional space ℝ³

$$H = {H_0} + {H_{\text{I}}} = \int {\psi {\text{*}}(x)} {\text{ }}\left( { - \frac{\Delta }{{2m}} - \mu } \right)\psi (x)dx + \frac{g}{2}\int {\psi {\text{*}}(x)\psi {\text{*}}(x\prime )\Phi (x{\text{ - }}x\prime )\psi (x\prime )\psi (x)dxdx\prime .} $$

((14.1))

Here, μ is a chemical potential, ψ*(x) and ψ(x) are operators of creation and annihilation independently of the type of statistics, and g is a coupling constant. Suppose that the frame of reference is chosen so that the lowest eigenvalue of the Hamiltonian is equal to zero

$$ H{\Phi _0} = 0.$$

(14.2)

The eigenvector Φ₀ that corresponds to the eigenvalue zero of the Hamiltonian H is called the ground state or “physical” vacuum.