Manuscript VII-B@1 · VII-B.tex · section · be632835

Consequences for second-order retarded products

Consequences for second-order retarded products · lines 1298–1407

VII-B@1#eq:four-pieces, VII-B@1#eq:conclusion

Suppose

V = κ V_{1} + κ^{2} V_{2} + O (κ^{3})

and a retarded map is normalized formally by

R_{V} (F) = F + R_{1} (V; F) + \frac{1}{2} R_{2} (V, V; F) + \dots .

Substituting the geometric expansion produces

[κ^{2}] R_{V} (B_{f}) = B_{f}^{(2)} + R_{1} (V_{1}; B_{f}^{(1)}) + R_{1} (V_{2}; B_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; B_{f}^{(0)}) .

This is a formal coefficient identity conditional on the existence of its terms.

The present result stops before the first mixed product:

R_{1} (V_{1}^{sc}; B_{f}^{(1)}) .

Its proposed geometric argument is not an element of the declared off-shell domain. No graph inventory or extension prescription for this product is therefore claimed here.

Item	Established status
Free multiplet and vertices	Specified, including ghosts and auxiliaries; see Section sec:domain and Appendix app:vertices.
Geometric bookkeeping	Equations (eq:B0)–(eq:B2) and (eq:classical-bookkeeping); no double counting.
Linear Cauchy adjoint	Computed exactly; its domain criterion is Theorem thm:linear-criterion.
First geometric composite	Excluded from the declared off-shell domain by Theorem thm:main.
Mixed and two-vertex products	Not constructed; the stated termwise off-shell route is blocked at its insertion-domain prerequisite.
Repairs	Linear smoothing and a free on-shell representative are proved. No nonlinear interacting repair is asserted.

The obstruction is not a failure of a diagonal extension theorem. The distribution $L^{*} a$ already exists, with compact support and finite transverse singular order. It fails the selected wavefront condition. Distribution-extension methods such as those of Brunetti and Fredenhagen BF, and local covariant time-ordered products of local Wick polynomials as constructed by Hollands and Wald HW, do not by themselves place this nonlocal sharp-Cauchy insertion in the declared domain.

Nor is the result an off-diagonal multiplication obstruction: no claim is made that a characteristic product of a coordinate kernel and a quantum two-point function is intrinsically undefined. Such a claim would require an additional analysis of the actual tensor-valued graph distributions.

The precise conclusion is therefore

sharp zero-Cauchy dressing + exact off-shell classical normalization + the polynomial microcausal domain ⟹ failure at B_{f}^{(1)} (f \neq = 0) .

A continuation must change at least one of these three requirements and prove that the new formulation preserves the intended observable in its stated sense.

No connected two-point response is declared in this paper. Accordingly, $B_{f}^{(1)} ⋆ B_{f^{'}}^{(1)}$ is not constructed. The boundary result neither computes a quantum response nor settles a later Ward or renormalization problem.

Displayed formulas in this passage

VII-B@1#eq:four-pieces · line 1309
$[κ^{2}] R_{V} (B_{f}) = B_{f}^{(2)} + R_{1} (V_{1}; B_{f}^{(1)}) + R_{1} (V_{2}; B_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; B_{f}^{(0)}) .$
VII-B@1#eq:conclusion · line 1384
$sharp zero-Cauchy dressing + exact off-shell classical normalization + the polynomial microcausal domain ⟹ failure at B_{f}^{(1)} (f \neq = 0) .$
unlabeled · line 1301
$V = κ V_{1} + κ^{2} V_{2} + O (κ^{3})$
unlabeled · line 1305
$R_{V} (F) = F + R_{1} (V; F) + \frac{1}{2} R_{2} (V, V; F) + \dots .$
unlabeled · line 1322
$R_{1} (V_{1}^{sc}; B_{f}^{(1)}) .$