Bosonic Bogoliubov transformation

Question

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
$hat{a}_p=u_phat{alpha}_{p}-v_phat{alpha}_{-p}^{dagger}$ and $hat{a}_{-p}^{dagger}=u_phat{alpha}_{-p}^{dagger}-v_phat{alpha}_{p}$.
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?

SuperCiocia · Accepted Answer

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.

Bosonic Bogoliubov transformation

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