Theorem: Complete classical quadratic kernel bridge

The quadratic observable is exactly

For compact perturbations this is a continuous quadratic functional. It has a symmetric distributional Hessian ${\mathcal{H}}_f$ satisfying

Its bilocal part is obtained by polarizing the second line of (eq:kernel-bridge). There are no additional classical quadratic terms for this coordinate prescription.

Proof. The last term of (eq:n2), divided by two, is $-{Y_1}^i{\partial }_i{n}_1$. Since ${Y_1}^i=-G_{\Delta }{B}^i\eta$, it is precisely the second line of (eq:kernel-bridge). All other terms are (eq:Llocal).

On any fixed compact input-support class, the kernel $|\mathbf{x}-\mathbf{y}{|}^{-1}$ is locally integrable. The differential operations $B,P$ have smooth coefficients on compact time intervals. Direct integral estimates give continuity in finite smooth seminorms.

Polarization gives the symmetric bilinear form

The local part is represented by derivatives of the diagonal delta distribution. The nonlocal part is represented by $G_{\Delta }(\mathbf{x}-\mathbf{y})\delta (t-u)$ with the displayed differential operations transferred to its two field slots. These are defined distributions before any quantum contraction. Their sum is ${\mathcal{H}}_f$.