Editing Bogoliubov transformation (section)

== Unified matrix description ==
Because Bogoliubov transformations are linear recombination of operators, it is more convenient and insightful to write them in terms of matrix transformations. If a pair of annihilators <math>(a , b)</math> transform as
:<math>
\begin{pmatrix}
\alpha\\
\beta
\end{pmatrix}
=
U
\begin{pmatrix}
a\\
b
\end{pmatrix}
</math>

where <math>U</math> is a <math>2\times2</math> matrix. Then naturally
:<math>
\begin{pmatrix}
\alpha^\dagger\\
\beta^\dagger
\end{pmatrix}
=
U^*
\begin{pmatrix}
a^\dagger\\
b^\dagger
\end{pmatrix}
</math>

For fermion operators, the requirement of [[commutation relations]] reflects in two requirements for the form of matrix <math>U</math>

:<math>
U=
\begin{pmatrix}
u & v\\
-v^* & u^*
\end{pmatrix}
</math>

and 

:<math>
|u|^2 + |v|^2 = 1
</math>

For boson operators, the [[commutation relations]] require
:<math>
U=
\begin{pmatrix}
u & v\\
v^* & u^*
\end{pmatrix}
</math>

and 

:<math>
|u|^2 - |v|^2 = 1
</math>

These conditions can be written uniformly as 
:<math>
U \Gamma_\pm U^\dagger = \Gamma_\pm
</math>
where
:<math>
\Gamma_\pm = 
\begin{pmatrix}
1 & 0\\
0 & \pm1
\end{pmatrix}
</math>
where <math>\Gamma_\pm</math> applies to fermions and bosons, respectively.

=== 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>
|}