States of scalar field on spacetimes with two isometries with timelike orbits

We define a pure state associated with every sufficiently regular state of linear quantum field theory. For a scalar field on a globally hyperbolic spacetime, the energy relative to a timelike vector field defines a quasifree state, whose associated pure state is a natural vacuum of the scalar field. If the energy state is defined relative to a globally timelike complete Killing field, its associated pure state is the well known `frequency splitting' stationary vacuum state. It is proved that such vacua corresponding to two commuting Killing fields are equal. It is conjectured that the vacua are equal even if the Killing fields do not commute. Implications for the particle interpretation of quantum field theory in curved spacetimes are discussed.