= t+x, U = t-x$. $\hat{a}_{1k}(\hat{a}_{2k})$, $\hat{a}_{1k}^{\dag} (\hat{a}_{2k}^{\dag} )$ are the corresponding annihilation and creation operators satisfying the bosonic commutation relations $$\begin{aligned}
[\hat{a}_{mk}, \hat{a}_{nk'}^{\dag}] = \delta_{mn} \delta({k-k'}), \,\,\, [\hat{a}_{mk}, \hat{a}_{nk'}] = [\hat{a}_{mk}^{\dag}, \hat{a}_{nk'}^{\dag}] = 0, \nonumber\end{aligned}$$ with $m,n = 1,2$. The Minkowski vacuum state $| 0_M \rangle$ is defined as $\hat{a}_{mk} | 0_M \rangle = 0$.

![Four wedges of $(1+1)$-dimensional Minkowski spacetime: $R, L, F$ and $P$. The right Rindler wedge ($R$) is causally disconnected to the left Rindler wedge ($L$). The Rindler coordinates $(\tau, \xi)$ only cover the $R$ wedge and $(\bar{\tau}, \bar{\xi})$ only cover the $L$ wedge. []{data-label="RindlerFrame"}](f1.png){width="8.0cm"}

As shown in Fig. \[RindlerFrame\], Minkowski spacetime can be divided into four wedges: $R, L, F$ and $P$. We introduce Rindler coordinates $(\tau, \xi)$ in the $R$ wedge and $(\bar{\tau}, \bar{\xi})$ in the $L$ wedge, $$\begin{aligned}
t &=& a^{-1} e^{a \xi} \sinh(a \tau), \,\,\, x = a^{-1} e^{a \xi} \cosh(a \tau),  \nonumber \\
t &=& -a^{-1} e^{a \bar{\xi}} \sinh(a \bar{\tau}), \,\,\, x = - a^{-1} e^{a \bar{\xi}} \cosh(a \bar{\tau}),\end{aligned}$$ where $\tau$ is the proper time of the uniformly accelerated observer with proper acceleration $a$ in the $R$ wedge. The metric is $g_{\mu\nu} = e^{2 a \xi} \text{diag}\{- 1, +1 \}$ 