Theorem: Exact fluctuation representation

For every residual state family,

Accordingly, every marginal fluctuation law is exactly the spectral law of $\Phi (v_j)$ in ${\rho }_{ϵ}$, and

On the common conjugated core,

For the displaced vacuum the fluctuation correlations are independent of $ϵ$:

When some $A_{ij}\ne 0$, their CCR algebra is noncommutative. If all $A_{ij}=0$, this selected sector is commuting and has an ordinary Gaussian fluctuation law.

Proof. Subtract the displacement constant in (eq:displacement) and divide by $\sqrt{ϵ}$. This proves the self-adjoint conjugation and hence the spectral statement. The unscaled Weyl relation gives (eq:fluctuation-Weyl); the unscaled core commutator gives (eq:fluctuation-CCR). The vacuum expectation was evaluated in Theorem thm:ordered.