which are invariant under Poincaré transformations. In curved space-time there is no global Poincaré symmetry.

In the following we consider a minimally coupled scalar field $\varphi$. Its Fourier modes in flat space are attached to the energy $\omega$ by $$\label{Energ}
{\mathrm{i}} \partial_t u_{k}= \omega u_k \quad \mbox{with} \quad \omega>0\; .$$ This uniqueness is lost when the coordiates are chosen arbitrarily. In curved space-time we have no distinguished time coordinate, thus no distinguished energy belonging to it, and no distinguished set of modes that can be related to the notion of particles. In general we have to deal with an arbitrary set of modes $(u_i)$ and ask for its relation to a different set $(v_i)$. The quantum field $\hat{\varphi}$ can then be expanded in both sets: $$\label{Phiua}
\hat{\varphi} = \sum_i (u_i \hat{a_i}+u_i^*\hat{a_i}^{\dag})\; , \quad
\hat{\varphi} = \sum_j (v_j \hat{b_j}+v_j^*\hat{b_j}^{\dag})\; .$$ The relations between these sets are given by the so-called Bogoliubov transformations [@Bogo] and have the form: $$\label{Bogu}
v_j = \sum_i (\alpha_{ji}u_i+\beta_{ji}u_i^*) \;, \quad
u_i = \sum_j (\alpha_{ji}^*v_j-\beta_{ji}v_j^*)$$ with $$\alpha_{jk} = (v_j,u_k) \; , \quad  \beta_{jk}=-(v_j,u_k^*) \;,$$ and the scalar product given by $$\label{skpr}
(u_i,v_j) =
-\mathrm{i}\int_{\Sigma}\left[u_i(\partial_{\mu} v_j^*) - 
                              (\partial_{\mu} u_i)v_j^*\right] 
           \sqrt{-g_\Sigma}\; \mathrm{d}\Sigma^{\mu}\;.$$ Here $\Sigma$ is a space-like hypersurface with volume 