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like in 2.32 we can see that:
[ B, B † ] = B B † −
( B † ) T ( B) T
†
(E.2)
leads to the desired matrix S. Unlike in matrices with scalar entries it’s not
allowed to simplify ( B † ) T ( B) T to ( B B † ) T , because the operators doesn’t com-
mute. This can be recognized by just looking on the first diagonal entries of
both expressions.
With the commutator defined it is possible to plug in the multi mode Bo-
goliubov transformation B = U P :
[ B, B † ] = U P P † U † −
= U P P † U † −
( P † U † ) T (U P ) T
†
(U † ) T
†
† T
(P ) P T T
U
= U P P † U † − (U T ) †
( P † ) T ( P ) T
†
U T
= U[ P , P † ]U †
The last line is only correct if U is assumed to be a real transformation (all
entries are real).
Note that in the second line instead of ( P † U † ) T the expression (U † ) T (P † ) T
can be written, because the multiplication of an operator from P † with a
scalar element of U commutes.
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