- #1

etotheipi

a_i' &= \int_{\Sigma_0} d^3 x \sqrt{h} n^a j_a(\psi_i', \Phi) \\

&= -i\int_{\Sigma_0} d^3 x \sqrt{h} n^a \big{(} \sum_j \left[ \bar{B}_{ij} \psi_j + \bar{A}_{ij} \bar{\psi}_{j} \right] (d\Phi)_a \\

&\quad\quad\quad\quad- \Phi \sum_j \left[ \bar{B}_{ij} (d\psi_j)_a + \psi_j (d\bar{B}_{ij})_a + \bar{A}_{ij} (d\bar{\psi}_j)_a + \bar{\psi}_j (d\bar{A}_{ij})_a \right] \big{)}

\end{align*}using the Bogoliubov transformation ##\bar{\psi}_i' = \sum_j \left[ \bar{B}_{ij} \psi_j + \bar{A}_{ij} \bar{\psi}_j \right]##. Is that right, and if so what's the next step toward the result? Thanks and sorry if I'm missing something obvious, I'm not really familiar with any of this subject.