= Solution
Consider a real <Klein-Gordon field> on a prescribed <globally hyperbolic spacetime>, obeying $(\Box-m^2)\phi=0$ with <metric signature> $(-,+,+,+)$. A curvature coupling can be included as $m^2\mapsto m^2+\xi R$. A <Cauchy hypersurface> $\Sigma$ and compactly supported smooth data $(\phi,n^a\nabla_a\phi)$ determine a unique solution; appropriate falloff can replace compact support. The real solution space has conserved <symplectic form>
$$
\Omega(\phi_1,\phi_2)=\int_\Sigma d\Sigma\,(\phi_1n^a\nabla_a\phi_2-\phi_2n^a\nabla_a\phi_1).
$$
Conservation follows by integrating the divergence-free current $\phi_1\nabla^a\phi_2-\phi_2\nabla^a\phi_1$, with no boundary flux. On complex solutions the conserved <Klein-Gordon inner product> is
$$
(u,v)_{KG}=i\int_\Sigma d\Sigma\,(u^*n^a\nabla_av-vn^a\nabla_au^*).
$$
It is indefinite on the full complex solution space.
Choose a complete positive-norm mode subspace, with modes $u_j$ satisfying
$$
(u_i,u_j)_{KG}=\delta_{ij},\quad (u_i^*,u_j^*)_{KG}=-\delta_{ij},\quad (u_i,u_j^*)_{KG}=0.
$$
Equivalently choose a compatible <complex structure on the Klein-Gordon solution space>. The mode labels may be continuous, in which case sums and Kronecker symbols become integrals and <Dirac delta functions>. Construct the one-particle <Hilbert space> from these modes and its <bosonic Fock space>. Promote the field to the operator-valued distribution
$$
\widehat\phi(x)=\sum_j\left(a_ju_j(x)+a_j^\dagger u_j^*(x)\right),\qquad [a_i,a_j^\dagger]=\delta_{ij},\quad [a_i,a_j]=[a_i^\dagger,a_j^\dagger]=0.
$$
For a foliation with spatial metric determinant $h$, the conjugate momentum density is $\widehat\pi=\sqrt h\,n^a\nabla_a\widehat\phi$. Mode completeness gives the equal-time <canonical commutation relations>
$$
[\widehat\phi(x),\widehat\pi(y)]=i\delta^3(x-y),\quad [\widehat\phi(x),\widehat\phi(y)]=[\widehat\pi(x),\widehat\pi(y)]=0.
$$
The <Fock vacuum> obeys $a_j|0\rangle=0$, and $a_j^\dagger a_j$ counts particles in the chosen mode. For local products and a renormalized <stress-energy tensor>, physically admissible states are further restricted by the <Hadamard condition>. The field algebra exists without a preferred <Fock vacuum>.
A different admissible mode splitting can mix positive and negative norms:
$$
v_i=\sum_j(\alpha_{ij}u_j+\beta_{ij}u_j^*),\qquad b_i=(v_i,\widehat\phi)_{KG}=\sum_j(\alpha_{ij}^*a_j-\beta_{ij}^*a_j^\dagger).
$$
Orthonormality imposes the <canonical identities for a bosonic Bogoliubov transformation>
$$
\alpha\alpha^\dagger-\beta\beta^\dagger=I,\qquad \alpha\beta^T=\beta\alpha^T.
$$
The same field then has
$$
\langle0_a|b_i^\dagger b_i|0_a\rangle=\sum_j|\beta_{ij}|^2.
$$
Without a preferred notion of <positive frequency>, a <nonstationary spacetime> supplies no distinguished mode splitting: \b[particle number and vacuum depend on the choice of modes, although the field equation and field algebra do not.] With infinitely many modes the <Bogoliubov transformation> need not be unitarily implementable; finite total mixing requires a <Hilbert-Schmidt operator> $\beta$.
For a stable <strictly stationary spacetime>, a chosen future globally timelike <Killing vector field> $K$ gives a preferred time translation. Choose modes with
$$
i\mathcal L_Ku_j=\omega_ju_j,\qquad \omega_j>0.
$$
When the corresponding conserved <Killing energy> is positive and the spectral problem has suitable boundary conditions and no problematic zero modes, this gives the preferred <vacuum state in a stationary spacetime> and particles relative to $K$. Positive-frequency mode mixing within that same subspace leaves the <Fock vacuum> unchanged. Rescaling $K$ by a positive constant changes the frequency units but not their sign.
\b[Stationarity alone, if it only means a Killing field timelike near infinity, is insufficient for a global unique particle interpretation.] In the <Kerr ergoregion>, $\partial_t$ is spacelike, so positive frequency relative to $t$ does not automatically select a positive-norm subspace throughout the geometry; <superradiance> illustrates the difficulty. Additional vacuum and boundary choices remain necessary. The customary stationary answer therefore assumes a suitable timelike stationary flow and a stable positive-energy quantization; it does not assert that every stationary black-hole extension has one globally preferred vacuum.
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