Past exam of the advanced quantum theory: path integrals and many-particle physics course of the University of Oxford / 2025 / 2 / b / Solution

Writing the Bogoliubov transformation componentwise, together with its Hermitian adjoint, givesThe full canonical anticommutation relations for $c,d$ implyEqual-type anticommutators vanish because $c^2=d^2=0$ and $\{c,d^{†}\}=\{c^{†},d\}=0$:For the mixed anticommutators, the only potentially nonzero terms cancel:while $\{a,b^{†}\}=0$ follows from $\{c,c\}=\{d^{†},d^{†}\}=\{c,d^{†}\}=0$. Taking the Hermitian adjoint supplies the remaining relations. Thus the necessary and sufficient condition isEquivalently, the matrix $R=\left(\begin{array}{cc}u& v\\ -v& u\end{array}\right)$ is an orthogonal matrix, $R^{\mathsf{T}}R=I$, giving a canonical rotation of two fermionic modes. Its inverse is therefore