Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

Local current before switching

Independent action variation and the plateau Noether identity › Local current before switching · lines 1727–1772

X-A-conv@1#eq:current, X-A-conv@1#eq:Noetherzero, X-A-conv@1#eq:Noetherone, X-A-conv@1#eq:Noethertwo

For the infinitesimal action (eq:infaction), naturality of the Hilbert and scalar densities gives

δ_{ξ} (ℓ + \partial_{μ} B^{μ}) = κ \partial_{μ} (ξ^{μ} (ℓ + \partial_{ν} B^{ν})) .

Consequently

δ_{ξ} ℓ = \partial_{μ} N^{μ}, N^{μ} = κ ξ^{μ} ℓ + κ ξ^{μ} \partial_{ν} B^{ν} - δ_{ξ} B^{μ} .

This formula explicitly defines a finite-order local current coefficient at every retained order. Negative-power current coefficients, if present, have identically zero divergence because the left side has no negative powers.

Write

n_{j}^{μ} = [N^{μ}]_{κ^{j}}, j = 0, 1, 2.

Coefficient comparison gives

δ_{ξ}^{(0)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{1} + δ_{ξ}^{(1)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{2} + δ_{ξ}^{(1)} ℓ_{1} = \partial_{μ} n_{0}^{μ}, = \partial_{μ} n_{1}^{μ}, = \partial_{μ} n_{2}^{μ} .

All $n_{j}$ depend locally on the fields and a finite jet of $ξ$ . They vanish outside the support of that jet.

Displayed formulas in this passage

X-A-conv@1#eq:current · line 1738
$δ_{ξ} ℓ = \partial_{μ} N^{μ}, N^{μ} = κ ξ^{μ} ℓ + κ ξ^{μ} \partial_{ν} B^{ν} - δ_{ξ} B^{μ} .$
X-A-conv@1#eq:Noetherzero · line 1759
$δ_{ξ}^{(0)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{1} + δ_{ξ}^{(1)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{2} + δ_{ξ}^{(1)} ℓ_{1} = \partial_{μ} n_{0}^{μ}, = \partial_{μ} n_{1}^{μ}, = \partial_{μ} n_{2}^{μ} .$
X-A-conv@1#eq:Noetherone · line 1759
$δ_{ξ}^{(0)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{1} + δ_{ξ}^{(1)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{2} + δ_{ξ}^{(1)} ℓ_{1} = \partial_{μ} n_{0}^{μ}, = \partial_{μ} n_{1}^{μ}, = \partial_{μ} n_{2}^{μ} .$
X-A-conv@1#eq:Noethertwo · line 1759
$δ_{ξ}^{(0)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{1} + δ_{ξ}^{(1)} ℓ_{0} δ_{ξ}^{(0)} ℓ_{2} + δ_{ξ}^{(1)} ℓ_{1} = \partial_{μ} n_{0}^{μ}, = \partial_{μ} n_{1}^{μ}, = \partial_{μ} n_{2}^{μ} .$
unlabeled · line 1731
$δ_{ξ} (ℓ + \partial_{μ} B^{μ}) = κ \partial_{μ} (ξ^{μ} (ℓ + \partial_{ν} B^{ν})) .$
unlabeled · line 1754
$n_{j}^{μ} = [N^{μ}]_{κ^{j}}, j = 0, 1, 2.$