Editing Bogoliubov transformation (section)

=== Diagonalizing a quadratic Hamiltonian using matrix description ===
Bogoliubov transformation lets us diagonalize a quadratic Hamiltonian
:<math>
\hat{H} =
\begin{pmatrix}
a^\dagger & b^\dagger
\end{pmatrix}
H
\begin{pmatrix}
a \\ b
\end{pmatrix}
</math>
by just diagonalizing the matrix <math>\Gamma_\pm H</math>. 
In the notations above, it is important to distinguish the operator <math>\hat{H}</math> and the numeric matrix <math>H</math>.
This fact can be seen by rewriting <math>\hat{H}</math> as
:<math>
\hat{H} =
\begin{pmatrix}
\alpha^\dagger & \beta^\dagger
\end{pmatrix}
\Gamma_\pm U (\Gamma_\pm H) U^{-1}
\begin{pmatrix}
\alpha \\ \beta
\end{pmatrix}
</math>

and <math>\Gamma_\pm U (\Gamma_\pm H) U^{-1}=D</math> if and only if <math>U</math> diagonalizes <math>\Gamma_\pm H</math>, i.e. <math>U (\Gamma_\pm H) U^{-1} = \Gamma_\pm D</math>.

Useful properties of Bogoliubov transformations are listed below.

{| class="wikitable"
|-
!         !! Boson !! Fermion
|-
| Transformation matrix || <math>U=\begin{pmatrix}u & v\\v^* & u^*\end{pmatrix}</math> || <math>U=\begin{pmatrix}u & v\\-v^* & u^*\end{pmatrix}</math>
|-
| Inverse transformation matrix || <math>U^{-1} = \begin{pmatrix}u^* & -v\\-v^* & u\end{pmatrix}</math> || <math>U^{-1}=\begin{pmatrix}u^* & -v\\v^* & u\end{pmatrix}</math>
|-
| Gamma || <math>\Gamma = \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}</math> ||  <math>\Gamma = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}</math>
|-
| Diagonalization || <math>U(\Gamma H) U^{-1} = \Gamma D </math> || <math>U H U^{-1} = D </math>
|}