Theorem: Constructed retarded extension

Equation (eq:referencePi) defines a real, tempered, future-causal tensor distribution. It agrees with (eq:Kubo) on $M\setminus \{0\}$ and has

Proof. Lemma lem:retardedexistence gives the reduced retarded distributions. Applying the constant-coefficient polynomial differential operators corresponding to ${\mathsf{Q}}_2,{\mathsf{Q}}_0$ preserves causal support and temperedness.

The retarded-minus-advanced difference is, by (eq:disc) and (eq:Wspec),

It is therefore a retarded splitting of the prescribed commutator. Away from the origin, causal support and the splitting identity give (eq:Kubo). Possible equal-time derivative terms from representing the stress tensor by polynomial differential operators are supported at the origin and are part of the extension prescription, not omitted nonlocal terms.

The boundary-value relation ${F_i}^{\mathrm{R}}(-p)=\overline{{F_i}^{\mathrm{R}}(p)}$, with real even tensor polynomials, proves reality.

Finally, the explicit form factors are bounded by $C\log (2+|p{|}_{\mathrm{E}})$. Thus

Under $x\mapsto \lambda x$, its pairing with a fixed Schwartz test is bounded by $C_{\psi }{\lambda }^{-8}(1+|\log \lambda |)$. Multiplication by ${\lambda }^{\omega }$ therefore gives zero as $\lambda \downarrow 0$ for every $\omega >8$. This is the claimed scaling-degree bound.