Theorem: Conditional causal subtraction

Suppose ${\mathsf{T}}_2$ is constructed on ${\mathcal{D}}_2$ with admitted coefficients and required descendants, classical multiplication, factorization, exact supplied old overlaps, and (eq:antitime)–(eq:time-star). Suppose the inherited free map has the corresponding admission and old-overlap properties. Define

Then:

1. $\hat{\mathsf{R}}(d)\in {\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, including required descendants;

2. its classical coefficient vanishes;

3. its supplied old unnormalized retarded overlaps agree exactly;

4. for $d=V\otimes O$, it vanishes if

   $${\mathrm{supp}}_{\mathrm{fun}}O\cap J^{+}({\mathrm{supp}}_{\mathrm{fun}}V)=\varnothing ;$$

5. $\hat{\mathsf{R}}(d{)}^{\ast }=\hat{\mathsf{R}}(\gamma d)$.

If both input maps satisfy the same named fixed-support continuity statement into a locally convex coefficient space, their difference satisfies that statement.

Proof. Admission and descendant admission follow from linearity of the admitted carrier. The classical coefficients of the two maps are both multiplication, so they cancel. Exact old overlap follows by subtracting the two exact old values.

The stated support condition is equivalent to

Thus $V\gtrsim O$, and factorization makes the two terms in (eq:Rhat) equal.

For conjugation, use (eq:free-star) and (eq:time-star):

Finally, addition and scalar multiplication are continuous in a locally convex vector space. No multiplication-continuity upgrade is used.