the same assertions are valid for the action of the groups $e^{itM_{0j}}$, $j = 2,\ldots,n-1$, on the suitable wedge algebras.

In Appendix \[sectionReehSchlieder\] it is proven that this analyticity entails that the theories we are considering here satisfy the Reeh–Schlieder property (cf. Lemma \[ReehSchlieder\]). So the vacuum vector $\Omega$ is cyclic for the algebra $\As(W)$, given any $W \in \Ws$, and, by the KMS-property, it is also separating for $\As(W)$ [@BratRob Corollary 5.3.9]. Hence, the Tomita–Takesaki modular theory is applicable to $(\As(W),\Omega)$, for every $W \in \Ws$ (cf. [@BR; @KadRing]). Let $J_{W_R}$ denote the modular conjugation and $\Delta_{W_R}^{it}$ the modular unitaries associated to the pair $(\As(W_R),\Omega)$. Since the adjoint action of the strongly continuous unitary group $e^{itM_{01}}$, $t \in \RR$, leaves the algebra $\As(W_R)$ invariant and satisfies the KMS condition, we must have $\Delta_{W_R}^{it} = e^{-i\beta tM_{01}}$, for all $t \in \RR$ [@KadRing Theorem 9.2.16]. Hence, $J_{W_R}$ is determined by the equation $$\label{jwr}
J_{W_R} A \Omega = e^{\, -(\beta/2) M_{01}} A^* \Omega  
\, , \   A \in {{\cal A}}(W_R) \, .$$

Unruh temperature
-----------------

The main task of this subsection is to determine the Unruh temperature $\beta^{-1}$ and specific properties of the operator $J_{W_R}$. To this end we shall adapt 