# Bosonic Bogoliubov transformation

In the Bogoliubov theory for the weakly interacting Bose gas, the Bogoliubov transformation from bosonic creation (annihilation) operators $$\hat{a}_p^{(\dagger)}$$ to the new set of creation (annihilation) operators $$\hat{\alpha}_p^{(\dagger)}$$ is defined via

$$\hat{a}_p=u_p\hat{\alpha}_{p}-v_p\hat{\alpha}_{-p}^{\dagger}$$ and $$\hat{a}_{-p}^{\dagger}=u_p\hat{\alpha}_{-p}^{\dagger}-v_p\hat{\alpha}_{p}$$.

I have looked various textbooks and lecture notes, and $$u_p$$ and $$v_p$$ are mostly assumed to be real. Why? Does this have some deeper meaning?

No deep meaning. The only requirement on $$u_p$$ and $$v_p$$ is their normalisation condition: $$|u_p|^2 + |v_p|^2 = 1.$$ For simplicity, and as usual in quantum mechanics, you just them to be real numbers.