Positive-frequency solution

Relative to a timelike Killing field $K$, a positive-frequency solution obeys $i{\mathcal{L}}_{K}u=\omega u$ with $\omega >0$. Such solutions define the one-particle space and annihilation operators of a stationary vacuum.

For $p^0=\sqrt{{\mathbf{p}}^2+m^2}>0$ in Minkowski spacetime, these modes obey $i{\partial }_t{\psi }_{\mathbf{p}}=p^0{\psi }_{\mathbf{p}}$ and have Klein-Gordon inner product $({\psi }_{\mathbf{p}},{\psi }_{\mathbf{q}})={\delta }^3(\mathbf{p}-\mathbf{q})$. Their conjugates have the opposite norm and are orthogonal to the positive-frequency modes. Plane waves are a generalized, delta-normalized basis; normalizable wave packets have positive norm $\int |a(\mathbf{p}){|}^2{d}^{3}p$. Completeness follows by applying the spatial Fourier transform to the Klein-Gordon equation, which gives the two branches $e^{\mp i{p}^{0}t}$ for each momentum.

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.