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In most papers where I've read about Rindler decomposition and the Unruh effect ( see for example [1] or [2]) they start by saying that they want to find the wavefunction of the vacuum state in the basis of states $\phi_L$ and $\phi_R$ living on the left and right sides of Minkowski space. They write the wavefunction in basis $\phi$ as

\begin{equation} \Psi\big(\phi \big)=\langle \phi |\Omega\rangle \propto\lim_{T\rightarrow \infty} \langle \phi_L \phi_R | e^{-HT}|\chi\rangle \\ \propto \int_{\phi(t_E=-\infty)=0}^{\phi(t_E=0)=\phi} \mathcal{D}\phi e^{-S_E} \end{equation} So far so good. Then, every review I found claims that you can make a change of variables such that instead of integrating from $\phi(t_E=-\infty)=0$ to $\phi(t_E=0)=\phi$ we integrate from $\phi(0)=\phi_L^*$ to $\phi(\pi)=\phi_R$ rotating with the boost operator $K_x$ in euclidean space. How does this work? I would think that the state $\phi$ that you want at the end is already $|\phi\rangle= |\phi_L \rangle |\phi_R\rangle$ so I don't see how you end with $\phi_L$ on one side of the boundary conditions and $\phi_R$ on the other one. Even more, it's not clear at all which coordinate change they are doing or why the euclidean action suddenly becomes the euclidean boost on x $K_x$. Once they asume that they write

\begin{equation} \Psi\big(\phi_L \phi_R \big)=\langle \phi_R | e^{-\pi K_x} \Theta | \phi_L\rangle \end{equation}

where $\Theta$ is the CPT operator. Once I trust the previous step where they "rotate", this result looks reasonable... but the rotation step seems very unclear to me.

[1] https://arxiv.org/abs/1409.1231

[2] https://arxiv.org/abs/1708.00748

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