In AdS/CFT, a particular duality is the correspondence between an eternal black hole in AdS spacetime (a large maximally extended AdS-Schwarzschild black hole) and the thermofield double state, \begin{equation} |TFD\rangle = \frac{1}{\sqrt{Z_{\beta}}}\sum_{i} e^{-\beta E_{i}/2} |E_{i}\rangle_{L} \otimes |E_{i}\rangle_{R} \end{equation} with $|E_{i}\rangle_{L}$ and $|E_{i}\rangle_{R}$ the Hamiltonian eigenstates for each CFT lining on the left (L) and right (R) boundary of the eternal black holes respectively, \begin{equation}\begin{aligned} H_{L}|E_{i}\rangle_{L} &= E_{i}|E_{i}\rangle_{L} \\ H_{R}|E_{i}\rangle_{R} &= E_{i}|E_{i}\rangle_{R} \end{aligned}\end{equation} where $H_{L}$ and $H_{R}$ are the Hamiltonians associated with the two copies of CFT. Each boundary Hamiltonian generates translations along the time directions $t_{L}$ and $t_{R}$ correspondingly, measured by boundary observers. These observers realize that subsystem they measure is thermal. The full boundary system then is evolved according to a ``united'' time $t$. The Hamiltonian of the full boundary system that generates translations along this time $t$ can be chosen to be either $H=H_{L}\otimes\mathbb{1}_{R}-\mathbb{1}_{L}\otimes H_{R}$ or $\tilde{H}=H_{L}\otimes\mathbb{1}_{R}+\mathbb{1}_{L}\otimes H_{R}$. Each choice has its own usefulness. Say that I choose $H$ in which case the thermofield double state is stationary since, \begin{equation} H|TFD\rangle = \frac{1}{\sqrt{Z_{\beta}}} \sum_{i} e^{-\beta E_{i}/2} \bigg[ \big(H_{L}|E_{i}\rangle_{L}\big)\otimes|E_{i}\rangle_{R} - |E_{i}\rangle_{L}\otimes\big(H_{R}|E_{i}\rangle_{R}\big)\bigg] = 0 \end{equation}

My question is, how is $t$ related to $t_{L}$ and $t_{R}$? That is, how can I explicitly write time evolution operator $\exp(-iHt)$ of the full system in terms of the time evolution operators $\exp(-iH_{L}t_{L})$ and $\exp(-iH_{R}t_{R})$ of the boundary subsystems?


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