Inherited state and the covariance convention

This input uses Em03-A's boundary-state prescription, :horizonoccupation, :quantumscatter, :numberdensity, and :packetlaw. Its bulk Hadamard identification remains the declared external Dappiaggi–Moretti–Pinamonti input DMP. Neither that global proof nor Em03-A's manuscript proofs are independently verified here.

Choose orthonormal $h_1,h_2\in {C_c}^{\infty }((0,\infty ))$, supported in a common compact band

Set $b_i=a(h_i)$ and $N_i={b_i}^{†}{b}_i$. Then $[b_i,{b_j}^{†}]={\delta }_{ij}$. Define

Thus

The transposition is important. Equation (eq:A) is not the formula for $⟨{b_i}^{†}{b}_j⟩$ with the same indices under the predecessor's test convention.

The matrix $A$ is positive semidefinite and satisfies

Figure fig:covariance fixes the index convention that the following characteristic function, passive transformation, and Wick formula all use. A complex off-diagonal entry is the control that exposes an accidental transpose.