Proposition: Conditional retarded-subtraction implication

Under Assumption ass:future, define on ${\mathcal{E}}_2$

This is an actual tensor map, equivalently a bilinear map on its declared pairs, with admitted coefficients and zero classical term. It has the exact declared old unnormalized retarded overlaps and satisfies

Furthermore,

If both terms have a proved continuity property at the same strength on the same named domain, their difference has that property.

Proof. Both summands are linear maps on the actual tensor domain ${\mathcal{E}}_2$, so their difference respects every actual relation. Its coefficients belong to the vector space ${\mathcal{P}}_{\Sigma }$, with the required descendants, by admission of the two coefficient systems. Equal classical terms cancel.

On an excluded causal pair, (eq:causalfuture) makes the difference zero. On ${\mathcal{E}}_{\mathrm{old}}$, the assumed old values and convention give the required old overlap.

For involution, use both order reversal and anti-time ordering:

The conjugate pair is in ${\mathcal{E}}_2$, by the stated domain stability.

Finally, subtraction is continuous in the coefficient topologies. Hence any common proved continuity statement is preserved.