= Solution
For a real free <scalar field> on a <globally hyperbolic spacetime>, take
$$
S=-\frac12\int d^4x\sqrt{-g}\left(\nabla_a\phi\nabla^a\phi+(m^2+\xi R)\phi^2\right).
$$
The <Klein-Gordon equation> has a well-posed initial-value problem on every <Cauchy hypersurface>. Its complex solution space carries the conserved <Klein-Gordon inner product>. Choose modes with
$$
(u_i,u_j)_{KG}=\delta_{ij},\qquad (u_i^*,u_j^*)_{KG}=-\delta_{ij},\qquad (u_i,u_j^*)_{KG}=0,
$$
and expand
$$
\widehat\phi=\sum_i(a_i u_i+a_i^\dagger u_i^*),\qquad [a_i,a_j^\dagger]=\delta_{ij}.
$$
The <canonical commutation relations> produce the <bosonic Fock space>, and $a_i|0\rangle=0$ defines the chosen vacuum.
The choice of <positive-frequency solutions> is extra structure. In a time-dependent geometry there is no preferred time translation, so different choices mix $u_i$ with $u_i^*$ and give the <vacuum ambiguity in a nonstationary spacetime>. In a <stationary spacetime>, a timelike <Killing vector field> defines positive frequency by $i\mathcal L_Ku_i=\omega_i u_i$, $\omega_i>0$, and hence the preferred <vacuum state in a stationary spacetime>.
If the spacetime is stationary in the remote past and future, define normalized in- and out-modes there. Their <Bogoliubov transformation> is
$$
u_i^{\rm out}=\sum_j(\alpha_{ij}u_j^{\rm in}+\beta_{ij}u_j^{{\rm in}*}).
$$
The expected out-particle number in the in-vacuum is
$$
\boxed{\langle0_{\rm in}|N_i^{\rm out}|0_{\rm in}\rangle=\sum_j|\beta_{ij}|^2}.
$$
Thus nonzero beta coefficients measure particle creation by the intervening time dependence.
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