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In my lecture today we learned about Bogoliubov transformations and the definition of unitarily implementable. We said a Bogoliubov map $$ B\colon h\oplus h'\to h\oplus h' $$ is unitarily implementable, iff there exists a unitary map $$U_B\colon \mathcal{F}_\pm \to \mathcal{F}_\pm$$ such that $$ U_B A_\pm(F) U_B^* =A_\pm(BF) \quad \forall F\in h\oplus h'.$$

We also discussed an example for an unitarily implementable Bogoliubov map, but to really understand the concept I would love to see a Bogoliubov map that is not unitarily implementable. Does anyone have an example of a bosonic and a fermionic Bogolubov transformation, that are not unitarily implementable?

Thank you very much!

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