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

Lemma: Exact current comparison

The explicit current contact calculation › A useful Noether representative · lines 869–895 · proof included or linked

Oai282-B@1#lem:super, Oai282-B@1#eq:superidentity

For every conformal Killing field,

j_{X}^{μ} - j_{N, X}^{μ} = \partial_{ρ} B_{X}^{ρ μ} .

Proof. Differentiate (eq:Bsuper). The conformal Killing equation combines the coefficients of $\partial Q$ into $- \frac{1}{2} σ_{X} \partial^{μ} Q$ . The last coefficient is

\frac{1}{12} (\partial^{μ} \partial_{ρ} X^{ρ} - □ X^{μ}) Q = \frac{1}{2} (\partial^{μ} σ_{X}) Q .

The second-derivative terms are

\frac{1}{6} (X^{μ} □ Q - X_{ν} \partial^{μ} \partial^{ν} Q) .

Since $: ϕ \partial^{μ} ϕ : = \frac{1}{2} \partial^{μ} Q$ , their sum is exactly $j_{X}^{μ} - j_{N, X}^{μ}$ .

Displayed formulas in this passage

Oai282-B@1#eq:superidentity · line 872
$j_{X}^{μ} - j_{N, X}^{μ} = \partial_{ρ} B_{X}^{ρ μ} .$
unlabeled · line 882
$\frac{1}{12} (\partial^{μ} \partial_{ρ} X^{ρ} - □ X^{μ}) Q = \frac{1}{2} (\partial^{μ} σ_{X}) Q .$
unlabeled · line 889
$\frac{1}{6} (X^{μ} □ Q - X_{ν} \partial^{μ} \partial^{ν} Q) .$

Nearby in this section

section: A useful Noether representative