in $R$ and is $g_{\mu\nu} = e^{2 a \bar{\xi}} \text{diag}\{- 1, +1 \}$ in $L$. It is obvious that the vector field $\partial_{\tau}~(\text{or}~ \partial_{\bar{\tau}})$ is the timelike Killing vector field of the spacetime [@Birrell82]. In the Rindler frame, the scalar field $\hat{\Phi}$ can be quantized as [@Fulling73; @Unruh76] $$\begin{aligned}
\hat{\Phi} = \int d \omega (\hat{b}_{1\omega}^R g_{1\omega}^R   + \hat{b}_{1\omega}^L g_{1\omega}^L  
  + \hat{b}_{2\omega}^R g_{2\omega}^R  + \hat{b}_{2\omega}^L g_{2\omega}^L +  \text{h.c.} ) $$ where the superscripts $``R"$ and $``L"$ represent modes and operators in the $R$ and $L$ wedge, respectively. The modes $g_{m\omega}^R~(g_{m\omega}^L)$ only have support in the $R$ ($L$) wedge, $$g^R_{1\omega}(v) = (4 \pi \omega)^{-1/2} e^{-i \omega v}, \,\,\, g^R_{2\omega}(u) = (4 \pi \omega)^{-1/2} e^{-i \omega u},$$ where $v = \tau + \xi$, $u = \tau - \xi$, and by replacing $v, u$ by $\bar{v} = - \bar{\tau} - \bar{\xi}$ and $\bar{u} = - \bar{\tau} + \bar{\xi}$ we obtain modes in the $L$ wedge. Note that we have used the prescription that $\partial_{\bar{\tau}}$ is past-directed. The commutation relations of the operators are $$\begin{aligned}
[\hat{b}_{m\omega}^R, \hat{b}_{n\omega'}^{R\dag} ] = \delta_{mn} \delta({\omega-\omega'}), \,\,\,\,
[\hat{b}_{m\omega}^L, \hat{b}_{n\omega'}^{L\dag} ] = \delta_{mn} \delta({\omega-\omega'}), \end{aligned}$$ with all others vanishing. The Rindler vacuum state $| 0_R \rangle$ is defined as $\hat{b}_{m\omega}^R | 