  We unify and generalize the notions of vacuum and amplitude in linear quantum
field theory in curved spacetime. Crucially, the generalized notion admits a
localization in spacetime regions and on hypersurfaces. The underlying concept
is that of a Lagrangian subspace of the space of complexified germs of
solutions of the equations of motion on hypersurfaces. Traditional vacua and
traditional amplitudes correspond to the special cases of definite and real
Lagrangian subspaces respectively. Further, we introduce both infinitesimal and
asymptotic methods for vacuum selection that involve a localized version of
Wick rotation. We provide examples from Klein-Gordon theory in settings
involving different types of regions and hypersurfaces to showcase generalized
vacua and the application of the proposed vacuum selection methods. A recurrent
theme is the occurrence of mixed vacua, where propagating solutions yield
definite Lagrangian subspaces and evanescent solutions yield real Lagrangian
subspaces. The examples cover Minkowski space, Rindler space, Euclidean space
and de Sitter space. A simple formula allows for the calculation of expectation
values for observables in the generalized vacua.
