Inverse transformation of Bogoliubov Transformation In the Quantum Field Thoery of Many-Body Systems book by Xiao-Gang Wen, he introduced the Bogoliubov's transformation in the first equation of page 74 as following:
$$
\alpha_k = u_ka_k + v_k a_{-k}^\dagger
$$
with $|u_k^2|-|v_k^2|=1$. Then he said in the next page that
$$
a_k = u_k^* \alpha_k - v_k^* \alpha_{-k}^\dagger
$$
But what I obtained for this inverse transformation is completely different. What I did was to consider
$$
\alpha_{-k}^\dagger = u_{-k}^*a_{-k}^\dagger + v_{-k}^*a_k
$$
Then, I obtained
$$
a_k = \frac{1}{u_ku_{-k}^*-v_kv_{-k}^*}\left(u_{-k}^*\alpha_k - v_k \alpha_{-k}^\dagger \right)
$$
Which is different from what he writes in the book.
 A: As far as I can see, your formula is correct. Let's transform this formula slightly.
From $[\alpha_k,\alpha_{-k}] = 0$, it follows
$$
\varphi_k \equiv \frac{v_k}{u_k} = \frac{v_{-k}}{u_{-k}}.
$$
Last equality together with $|u_k|^2 - |v_k|^2 = 1$ and $|u_{-k}|^2-|v_{-k}|^2 = 1$ leads to equalities
$$
u_k = \frac{e^{i\gamma_k}}{\sqrt{1-|\varphi_k|^2}},
\ v_k = \frac{e^{i\gamma_k}\varphi_k}{\sqrt{1-|\varphi_k|^2}},
\ u_{-k} = \frac{e^{i\gamma_{-k}}}{\sqrt{1-|\varphi_k|^2}},
\ v_{-k} = \frac{e^{i\gamma_{-k}}\varphi_k}{\sqrt{1-|\varphi_k|^2}},
$$
where $\gamma_k$, $\gamma_{-k}$ are real numbers. It is straightforward to obtain relations
$$
u^*_{-k}u_k - v^*_{-k}v_k = e^{i(\gamma_k - \gamma_{-k})},
$$
$$
\frac{u^*_{-k}}{u^*_{-k}u_k - v^*_{-k}v_k} = u^*_k,
\ \frac{v_k}{u^*_{-k}u_k - v^*_{-k}v_k} = v_{-k}.
$$
Hence, we've transformed the initial formula to the following form
$$
a_k = u^*_k\alpha_k - v_{-k}\alpha_{-k}^\dagger. \quad (*)
$$
The last formula is still different from the book's one. So I assume there is a misprint in the book.
