Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 1 / c / Solution

For real even ${\theta }_k$, the Bogoliubov transformation givesEquivalently, its Nambu matrix obeys $T_k{\sigma }_z{T}_k^{†}={\sigma }_z$. The bosonic canonical metric is preserved; $T$ need not be unitary in the ordinary positive-definite metric.