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

Descendants and Wick-block contacts

The explicit current contact calculation › Descendants and Wick-block contacts · lines 1005–1046

Oai282-B@1#eq:jetcontact, Oai282-B@1#eq:blockC

For an elementary descendant, define

C_{X, α}^{μ} (x, y) = \partial_{y}^{α} C_{X}^{μ} (x, y) .

Then

\partial_{μ}^{x} C_{X, α}^{μ} (x, y) = \partial_{y}^{α} [δ^{(4)} (x - y) v_{X} ϕ (y)] .

Thus derivative-of-delta contacts are retained:

\partial_{y}^{α} [δ (x - y) v_{X} ϕ (y)] = β \leq α \sum (β α) (\partial_{y}^{α - β} δ (x - y)) \partial_{y}^{β} v_{X} ϕ (y) .

For a Wick block $B_{α} = : \prod_{a} \partial^{α_{a}} ϕ :$ , the single-block insertion is the sum over its participating factors:

C_{X, B}^{μ} (x, y) = a = 1 \sum^{k} : C_{X, α_{a}}^{μ} (x, y) b \neq = a \prod \partial^{α_{b}} ϕ (y) : .

Here the colon orders all remaining quantum factors; it does not normal-order the numerical retarded kernel.

Displayed formulas in this passage

Oai282-B@1#eq:jetcontact · line 1012
$\partial_{μ}^{x} C_{X, α}^{μ} (x, y) = \partial_{y}^{α} [δ^{(4)} (x - y) v_{X} ϕ (y)] .$
Oai282-B@1#eq:blockC · line 1034
$C_{X, B}^{μ} (x, y) = a = 1 \sum^{k} : C_{X, α_{a}}^{μ} (x, y) b \neq = a \prod \partial^{α_{b}} ϕ (y) : .$
unlabeled · line 1008
$C_{X, α}^{μ} (x, y) = \partial_{y}^{α} C_{X}^{μ} (x, y) .$
unlabeled · line 1020
$\partial_{y}^{α} [δ (x - y) v_{X} ϕ (y)] = β \leq α \sum (β α) (\partial_{y}^{α - β} δ (x - y)) \partial_{y}^{β} v_{X} ϕ (y) .$