Vacuum state in a stationary spacetime

In a strictly stationary spacetime, the timelike Killing flow gives a preferred positive-frequency splitting. The corresponding vacuum is annihilated by every annihilation operator associated with a positive-frequency mode.

For a stable free Klein-Gordon field with a complete timelike Killing flow and a positive self-adjoint one-particle energy generator, its positive spectral subspace selects positive-frequency solutions of positive Klein-Gordon inner product. Their annihilation operators define the ground-state Fock vacuum. Degenerate modes can be mixed by a unitary matrix without changing this splitting or vacuum. Merely being stationary near infinity is insufficient if the Killing field becomes spacelike in an ergoregion of a stationary spacetime; zero modes, instability or additional boundary conditions also require separate treatment. The choice of a ground state does not prevent other states, such as thermal states, from using the same particle operators.

// Target: statistical-physics.bigb

This is the symmetric-particle construction of Fock space.

For one-particle Hilbert space ${\mathcal{H}}_1$, the bosonic Fock space isCreation operators add one symmetrized particle and annihilation operators remove one.