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

A useful Noether representative

The explicit current contact calculation › A useful Noether representative · lines 838–868

Oai282-B@1#eq:jN, Oai282-B@1#eq:Bsuper

Define

j_{N, X}^{μ} = T^{μν} X_{ν} + σ_{X} : ϕ \partial^{μ} ϕ : - \frac{1}{2} (\partial^{μ} σ_{X}) Q .

Before imposing the free equation, the corresponding polynomial calculation gives

\partial_{μ} j_{N, X}^{μ} = : (□ ϕ) d_{X} ϕ : .

The identity uses (eq:CK) and (eq:CKaux).

Define the antisymmetric superpotential

B_{X}^{ρ μ} = \frac{1}{6} (X^{μ} \partial^{ρ} Q - X^{ρ} \partial^{μ} Q) + \frac{1}{12} Q (\partial^{μ} X^{ρ} - \partial^{ρ} X^{μ}) .

Displayed formulas in this passage

Oai282-B@1#eq:jN · line 841
$j_{N, X}^{μ} = T^{μν} X_{ν} + σ_{X} : ϕ \partial^{μ} ϕ : - \frac{1}{2} (\partial^{μ} σ_{X}) Q .$
Oai282-B@1#eq:Bsuper · line 859
$B_{X}^{ρ μ} = \frac{1}{6} (X^{μ} \partial^{ρ} Q - X^{ρ} \partial^{μ} Q) + \frac{1}{12} Q (\partial^{μ} X^{ρ} - \partial^{ρ} X^{μ}) .$
unlabeled · line 851
$\partial_{μ} j_{N, X}^{μ} = : (□ ϕ) d_{X} ϕ : .$