Representation and common domain

A useful fixed purified realization is the doubled Fock space with real-linear map

It satisfies

Let $\Omega$ be its vacuum. Its physical Weyl subalgebra has exactly the prescribed characteristic function. No global density in the undoubled stationary radiation Fock space is assumed.

Write $N_{\mathrm{d}}$ for doubled Fock particle number. Segal fields are essentially self-adjoint on finite-particle vectors, and $\mathcal{D}((N_{\mathrm{d}}+1{)}^{1/2})$ is their common form domain. Weyl displacements preserve this domain.