Proposition: Consistent quadratic CTP coefficient

The four kernels satisfy

The quadratic connected CTP cumulant, with $V_H=-g\mathsf{T}(H)$, is

In particular ${{\Gamma }_{\mathrm{CTP}}}^{[2]}[H,H]=0$. For real $H_{\mathrm{d}}$, the imaginary quadratic form is nonnegative.

The construction is restricted to source degree two. The four kernels are not a closed-time-path instrument, a measurement channel, a classical record, or a decoder.

Proof. Equation (eq:disc) gives

The same expression is ${\hat{D}}^{>}+{\hat{D}}^{<}$. Subtracting the negative-energy or positive-energy Wightman component from $D^{++}$ yields the retarded or advanced boundary prescription. The contacts have been transported with opposite signs on the two diagonal branches, so the identities retain them.

The vacuum source-degree-two cumulant has the displayed two self-branch terms and the two cross-branch terms. Substitute $H_{\pm }=H_{\mathrm{c}}\pm H_{\mathrm{d}}/2$. Use the first identity to cancel the $H_{\mathrm{c}}{H}_{\mathrm{c}}$ term and $D^{>}(x-y)=D^{<}(y-x)$ with exchanged tensor pairs. The mixed term is $g^2{H}_{\mathrm{d}}{\Pi }^{\mathrm{R}}{H}_{\mathrm{c}}$; the remaining difference-source term is $\mathrm{i}{g}^2{H}_{\mathrm{d}}N{H}_{\mathrm{d}}/(2\hslash )$.

For a real test, $H_{\mathrm{d}}N{H}_{\mathrm{d}}$ equals a nonnegative vacuum norm, proving the sign. No selective maps or quantum output registers occur in this calculation.