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

Normal-ordering compatibility

Compact-current implementation on the complete field algebra › Normal-ordering compatibility · lines 1244–1279

Oai282-B@1#eq:prolong, Oai282-B@1#eq:jetaction

Define the prolonged evolutionary action by

v_{X} (\partial^{α} ϕ) = \partial^{α} (d_{X} ϕ), v_{X} B_{α} = a = 1 \sum^{k} : \partial^{α_{a}} (d_{X} ϕ) b \neq = a \prod \partial^{α_{b}} ϕ : .

The explicit jet formula is

\partial^{α} (d_{X} ϕ) = X^{μ} \partial_{μ} \partial^{α} ϕ + 0 < β \leq α \sum (β α) (\partial^{β} X^{μ}) \partial_{μ} \partial^{α - β} ϕ + β \leq α \sum (β α) (\partial^{β} σ_{X}) \partial^{α - β} ϕ .

In particular, descendants are not silently treated as primaries.

Displayed formulas in this passage

Oai282-B@1#eq:prolong · line 1247
$v_{X} (\partial^{α} ϕ) = \partial^{α} (d_{X} ϕ), v_{X} B_{α} = a = 1 \sum^{k} : \partial^{α_{a}} (d_{X} ϕ) b \neq = a \prod \partial^{α_{b}} ϕ : .$
Oai282-B@1#eq:jetaction · line 1262
$\partial^{α} (d_{X} ϕ) = X^{μ} \partial_{μ} \partial^{α} ϕ + 0 < β \leq α \sum (β α) (\partial^{β} X^{μ}) \partial_{μ} \partial^{α - β} ϕ + β \leq α \sum (β α) (\partial^{β} σ_{X}) \partial^{α - β} ϕ .$