Fixing the complete environment

A complement cannot be defined by tracing selected modes from an unpurified mixed packet marginal. We fix the following representation.

In the doubled realization of Section sec:dynamics, the regular selected finite-mode CCR representation has a tensor factorization

The selected packet algebra acts on the first factor. One way to obtain this factorization is to complete its finitely many canonical quadratures to a real symplectic basis in their finite-dimensional doubled one-particle span, implement that finite canonical transformation, and leave the orthogonal doubled Fock space unchanged. This is the finite CCR multiplicity construction.

We fix this factorization once. The vector $\Omega$ is its fixed purification of the selected background. The environment contains:

1. unselected participating radiation;

2. all other outgoing-infinity modes;

3. the future-horizon field algebra;

4. the purification degrees of freedom of the incoming thermal horizon sector.

These physical field subalgebras need not be independent tensor factors in the chosen background. They are not discarded from the environmental representation.

Define the Stinespring isometry

Its accessible output is the selected packet factor, and its environment is $S\otimes {\mathcal{H}}_{\mathrm{env}}$.