TBBT Physics Bowl

In one of the episodes of The Big Bang Theory, the gang enters into the department physics bowl. Near the end, Sheldon was stunned by the following Feynman diagram problem:

Considering that this is one of the basic diagram for QED, it would be impossible for the character to not even recognize the problem. Anyway, the janitor then answered that the correct answer is \( -8 \pi \alpha \). It is incorrect. Well, at least not without additional conditions not specified by the problem.

Two bands interaction: self-energy and effective low energy Hamiltonian.

Consider the following model of two-bands Hamiltonian with interaction between the bands:

$$ H = \left( \begin{array}{cc} H_{11} &H_{12} \\ H_{12}^{\dag} & H_{22} \end{array} \right)\,, $$

where $H_{11}$ and $H_{22}$ are some low and high energy bands that we know how to diagonalize and $H_{12}$ is some interaction between them. Since we know the spectrum of $H_{11}$, we can write down the GF. Here it is:

$$ G_{11}^{(0)} = \frac{1}{H_{11} - \epsilon^{0}} $$