element $\mathrm{d}\Sigma$, $\mathrm{d}\Sigma^{\mu}=n^{\mu}\mathrm{d}\Sigma$, and $n^{\mu}$ is a normal vector of $\Sigma$ with norm $n^{\mu}n_{\mu}=1$.

The creation and annihilation operators (\[Phiua\]) depend on the modes, and so does the vacuum state defined with respect to them. The vacuum state $|0_b\rangle$ belongs to $\hat{b}_i$ and is defined by $\hat{b}_i|0_b\rangle=0$. In general, $|0_b\rangle$ is not identical with $|0_a\rangle$, for which we have $\hat{a}_i|0_a\rangle=0$. It follows that the vacuum expectation value of the number density operator $\hat{a}_i^{\dag} \hat{a}_i$ must not necessarily vanish when applied to the vacuum $|0_b\rangle$, $$\label{Vakew}
\langle0_b|\hat{a}_i^{\dag} \hat{a}_i|0_b\rangle=\sum_j |\beta_{ji}|^2\;.$$ This operator has a well defined meaning as the number (mode) density operator only if the modes used for the expansion can be put in a suitable relation to the known modes of Minkowski space-time to render an interpretation possible. This is possible if we are interested in the change of vacuum between two asymptotically flat regions of space-time.

If the eigenvalues are not discrete $(i \to \vec{k})$, we have to deal with an integral that may have divergences. These divergences caused by the limit from periodic boundary conditions with discrete spectrum to an infinite volume ${\bf{V}}=\langle 0|0\rangle$ can be cured by normalization. This may be accomplished by dividing through the same 