Manuscript Oai282-B@1 · Oai282-B.tex · section · f2956de5

The spacelike-collar contract

Compact-current implementation on the complete field algebra › The spacelike-collar contract · lines 1097–1127

Oai282-B@1#eq:q

Let $K_{0} \subset M$ be a compact insertion support, and $I \subset R$ a compact time interval. Choose

a \in C_{c}^{\infty} (I; R), \int a (t) d t = 1.

Choose $χ \in C_{c}^{\infty} (R^{3}; R)$ , equal to one on a neighborhood of

{x : (t, x) \in J (K_{0}) for some t \in supp a} .

This set is bounded. We also require the support of $\nabla χ$ , throughout $supp a$ , to be strictly spacelike separated from $K_{0}$ . Such a choice is obtained by making the one-region of $χ$ larger than the compact causal shadows.

Define the actual spacetime-smeared operator

q_{X} [a, χ] = \int a (t) χ (x) j_{X}^{0} (t, x) d t d^{3} x .

It acts on $D_{E}$ by Lemma lem:wickest.

Displayed formulas in this passage

Oai282-B@1#eq:q · line 1119
$q_{X} [a, χ] = \int a (t) χ (x) j_{X}^{0} (t, x) d t d^{3} x .$
unlabeled · line 1101
$a \in C_{c}^{\infty} (I; R), \int a (t) d t = 1.$
unlabeled · line 1108
${x : (t, x) \in J (K_{0}) for some t \in supp a} .$

Nearby in this section

remark: Remark: No unsupported sharp-time operator is used