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Choose the unprimed modes as the inertial positive-frequency modes in the far past, and the primed modes as those in the far future. In the in-vacuum and out-vacuum construction, $|0_{\mathrm{in}}⟩$ is annihilated by every $a_j$. The number of late-time quanta in mode $i$ is measured by $N_i^{\prime }=a_i^{\prime }{}^{†}{a}_i^{\prime }$. Expanding using the Bogoliubov transformation, the only surviving vacuum contraction is $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$. Thus the particle number from Bogoliubov coefficients isIf $B\ne 0$, at least one row has positive norm and late-time observers detect particles. If $B=0$, each $a_i^{\prime }$ is a linear combination of past annihilation operators and annihilates the same state, so all these occupation numbers vanish. The intervening nonstationary geometry can mix positive and negative frequencies, realizing vacuum ambiguity in a nonstationary spacetime even though the asymptotic inertial choices are unambiguous.

For infinitely many modes, the expressions are understood with suitable convergence or wave-packet regularization. The total expected number is ${\sum }_{ij}|B_{ij}{|}^2$, finite precisely when $B$ is a Hilbert-Schmidt operator in the specified orthonormal mode spaces. A divergent sum is not a finite occupation number in a common vacuum representation; it signals the extra infinite-dimensional issue rather than changing the finite-mode calculation.

Choose complete orthonormal sets of complex Klein-Gordon solutions $\{u_i^{\mathrm{in}}\}$ and $\{u_i^{\mathrm{out}}\}$ 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:The real quantum field has either mode expansion of a free field

Completeness relates the mode bases by the Bogoliubov transformationand hencePreservation of the canonical commutation relations requiresThe 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.