Lemma: Reference extension

The distribution (eq:reference-response) is real, invariant under proper orthochronous Lorentz transformations, retarded, and extends ${\mathcal{R}}^{\circ }$. Its scaling degree is four. For every compact $K\subset M$, there is a finite constant $C_{K,\mu }$ such that

where $p_{j,K}$ is the maximum of the derivatives through order $j$ on $K$.

Proof. Logarithmic singularities are locally integrable transverse to the nonzero cone. At the tip, radial integration also gives a finite integral of a logarithm. Reality, Lorentz invariance, and retarded support follow from the defining function and the operator ${\square }^2$.

We verify the constant using the retarded Riesz family. In dimension four its normalization is

on its locally integrable domain, continued holomorphically as a distribution. The identities

are the standard Riesz identities BGP (Section 1.2).

On the punctured future cone, put $Y_{\beta }(u)=u_{+}^{\beta -1}/\Gamma (\beta )$, continued in $\beta$. Since $Y_{-1}={\delta }^{\prime }$ and $1/\Gamma (\alpha /2)=\alpha /2+O({\alpha }^2)$,

On the other hand,

Differentiate ${\square }^2{\mathsf{R}}_{\alpha +4}={\mathsf{R}}_{\alpha }$ at zero. The derivative of $C$ contributes only a multiple of ${\delta }_0$, and the logarithmic term contributes

It follows that (eq:reference-response) restricts to $\hslash \Theta (t){\delta }^{\prime }(s)/(4{\pi }^3)$, as required.

For the estimate, weak differentiation gives

The absolute logarithmic integral on $K$ is finite, and $\left|{\square }^{2}F\right|\le 16p_{4,K}(F)$.

Finally, positive dilation of this formula gives the exact logarithmic scaling identity proved below. It implies scaling degree at most four. The nonzero punctured homogeneous restriction makes the degree exactly four.