Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2016 / Paper 1 / 1 / e

Write $u_k=\cosh {\theta }_k$, $v_k=\sinh {\theta }_k$, taking real even ${\theta }_k$. The transformation givesThe second equality uses $u_{-k}=u_k$ and $v_{-k}=v_k$, while the first uses $u_k^2-v_k^2=1$. The creation-creation commutator vanishes by conjugation. Therefore the $b$ modes are canonical bosons. In the paired basis,Its determinant is one and its inverse reverses the signs of the off-diagonal entries. Hence $\boxed{a_k=u_k{b}_k-v_k{b}_{-k}^{†}}$. At self-inverse momenta $k=0,\pi$ this is the corresponding one-mode squeezing transformation, still canonical for finite ${\theta }_k$.