= Solution
Choose complete orthonormal sets of complex Klein-Gordon solutions $\{u_i^{\rm in}\}$ and $\{u_i^{\rm out}\}$ that have positive frequency with respect to the asymptotic timelike Killing fields in the remote past and future. Their normalization uses the conserved <Klein-Gordon inner product>:
$$
(u_i,u_j)_{KG}=\delta_{ij},
\qquad
(u_i^*,u_j^*)_{KG}=-\delta_{ij},
\qquad
(u_i,u_j^*)_{KG}=0.
$$
The real quantum field has either <mode expansion of a free field>
$$
\phi=\sum_i(a_i^{\rm in}u_i^{\rm in}
+a_i^{{\rm in}\dagger}u_i^{{\rm in}*})
=\sum_i(a_i^{\rm out}u_i^{\rm out}
+a_i^{{\rm out}\dagger}u_i^{{\rm out}*}).
$$
Completeness relates the mode bases by the <Bogoliubov transformation>
$$
u_i^{\rm out}=\sum_j
(\alpha_{ij}u_j^{\rm in}+\beta_{ij}u_j^{{\rm in}*}),
$$
and hence
$$
a_i^{\rm out}=\sum_j
(\alpha_{ij}^*a_j^{\rm in}-\beta_{ij}^*a_j^{{\rm in}\dagger}).
$$
Preservation of the <canonical commutation relations> requires
$$
\alpha\alpha^\dagger-\beta\beta^\dagger=I,
\qquad
\alpha\beta^T=\beta\alpha^T.
$$
The past and future notions of positive frequency therefore define the generally different <in-vacuum and out-vacuum>. Nonzero $\beta$ means that the in-vacuum is a squeezed state containing out-particles.
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