Manuscript XIII-B@1 · XIII-B.tex · section · b8a5c0f5

B4: causal support and classical evaluation

B4: causal support and classical evaluation · lines 1076–1110

XIII-B@1#eq:retarded-attachment, XIII-B@1#eq:functional-support

The established tree attachment has support

supp G_{act}^{R} (x, y) \subset {(x, y) : x \in J^{+} (y)} .

Thus the interaction attachment can affect the contracted observable slot only to its future. If the entire interaction support lies later than the entire functional support of $O$ , the tree datum is zero. The analogous complete binary statement is conditional on the extension normalization in Theorem thm:conditional.

There is also the valid compact output bound

supp_{func} R_{1}^{(0)} \subseteq supp_{func} V_{g} \cup supp_{func} O .

The uncontracted slots are projections of the two derivative-kernel supports.

Equation (eq:retarded-attachment) does not imply

supp_{func} R_{1}^{(0)} \subseteq J^{+} (supp_{func} V_{g}) .

An untouched metric slot of the nonlocal observable may lie earlier than the interaction while another slot receives the retarded attachment. A claim about every functional-support variable requires its own proof. We therefore use the attachment support condition, not the unsupported stronger inclusion.

Displayed formulas in this passage

XIII-B@1#eq:retarded-attachment · line 1080
$supp G_{act}^{R} (x, y) \subset {(x, y) : x \in J^{+} (y)} .$
XIII-B@1#eq:functional-support · line 1091
$supp_{func} R_{1}^{(0)} \subseteq supp_{func} V_{g} \cup supp_{func} O .$
unlabeled · line 1101
$supp_{func} R_{1}^{(0)} \subseteq J^{+} (supp_{func} V_{g}) .$