In the Jerusalem Lectures on Black Holes section 3.3 the author considers a QFT in Minkowski space. He then picks out a space coordiante, say $x$, and divides the Hilbert space $H$ of the QFT in two factors $H=H_L\otimes H_R$, where $H_L$ is acted on by fields $\phi(x)$ with $x<0$ and $H_R$ analog $x>R$. After some calculations, he arrives in equation $3.22$ at the picture where the vacuum state of the QFT appears as an entangled state \begin{align} \left|\Omega\right>= \frac{1}{\sqrt{Z}}\sum_ne^{-\pi \omega_n}\left|n_L\right>\otimes \left|n_R\right> \end{align} which is just a thermal state at temperature $T=(2\pi)^{-1}$. Here $\left|n_{L/R}\right>$ are eigenstates of the boost operator $K_x$, that mixes $x$ and $t$ coordinates.
Now de Sitter space is pretty similar to Rindler space, where the two static patches take the role of the rindler wedges $x>0$ and $x<0$. My question is therefore, if there is also a similar decomposition of the Bunch-Davies vacuum state of a QFT in de Sitter space in eigenstates of some operator in the left and right static patches? Also I am not really sure what the $\omega_n$ in this expansion are. I would be really thankful if someone has some literature on this on hand, since I can't find anything on this on the internet.
