Green's Functions

Propagator theory is based on the Green's function method of solving inhomogeneous differential equations. We explain the method in terms of a single example.

Suppose we wish to solve Poisson's equation

$$\nabla^2\phi(\vec{x}) = -\rho(\vec{x})$$ (6.1)

for a known charge distribution $\rho(\vec{x})$, subject to some boundary conditions. It is easier to first solve the ``unit source'' problem

$$\nabla_{\vec{x}}^2 G(\vec{x};\vec{x}^\prime) = -\delta^{(3)}(\vec{x}-\vec{x}^\prime)$$ (6.2)

where $G(\vec{x};\vec{x}^\prime)$ is the potential at $\vec{x}$ due to a unit source at $\vec{x}^\prime$. We then move this source over the charge distribution and accumulate the total potential at $\vec{x}$ from all possible volume elements $d^3x^\prime$:

$$\phi(\vec{x}) = \int G(\vec{x};\vec{x}^\prime) \rho(\vec{x}^\prime) d^3x^\prime .$$ (6.3)

We can check directly that $\phi$ is the desired solution.

$$\nabla_{\vec{x}}^2\phi(\vec{x}) = \int \nabla_{\vec{x}}^2 G(\vec{x};\vec{x}^\prime) \rho(\vec{x}^\prime) d^3x^\prime$$ (6.4)

$$= -\int \delta^3(\vec{x}-\vec{x}^\prime) \rho(\vec{x}^\prime) d^3x^\prime$$

$$= -\rho(\vec{x}) .$$

In the case of the electron propagator, $\psi$ appears on both sides of the solution, and so an iterative perturbation series-solution in powers of $e$ is required.