Past exam of the advanced quantum theory: path integrals and many-particle physics course of the University of Oxford / 2024 / 2 / b / Solution

Taking the Hermitian adjoint gives ${\phi }_{\mathbf{k}}=c_k{\psi }_{\mathbf{k}}+s_k{\psi }_{-\mathbf{k}}^{†}$. For nonzero wavevectors, the canonical commutation relations implyWhen the Kronecker delta is nonzero, $k=q$, so the hyperbolic Pythagorean identity $c_k^2-s_k^2=1$ givesFor the two annihilation operators, the only potentially nonzero terms areHere $\mathbf{q}=-\mathbf{k}$ implies equal magnitudes and hence equal $c,s$, because ${\theta }_k$ depends only on $|\mathbf{k}|$. ThusThe corresponding commutators of two creation operators vanish by the Hermitian adjoint. This verifies that the Bogoliubov transformation preserves the full bosonic canonical commutation relations.