Manuscript V-B@1 · V-B-v1.tex · theorem · 3fadc1ad

Theorem: Complete defect

Raw current, surface term, and complete Ward defect · lines 489–529 · proof included or linked

V-B@1#thm:defect, V-B@1#eq:H, V-B@1#eq:Hab, V-B@1#eq:gaugedefect

Let $H = K^{*} J_{raw}$ . Then

H = γ + δ (t) a + δ^{'} (t) b_{0},

where, componentwise,

γ^{ν} a^{ν} b_{0}^{ν} = f \partial^{ν} ϕ, = u ˙_{0} \partial^{ν} ϕ_{0} - u_{0} \partial_{t} \partial^{ν} ϕ ∣_{0}, = u_{0} \partial^{ν} ϕ_{0} .

For every smooth descriptor,

B_{f} (K ξ, ϕ) = ⟨ H, ξ ⟩ .

Proof. For a scalar distribution $v$ , the product rule gives

2 \partial_{μ} (\partial^{(μ} v \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (v \partial^{α} ϕ)) = (P v) \partial^{ν} ϕ,

using $P ϕ = 0$ . Also

P (Θ u) = - f - δ (t) u ˙_{0} - δ^{'} (t) u_{0} .

Multiplying the last term by $\partial^{ν} ϕ (t)$ and using $c (t) δ^{'} = c (0) δ^{'} - c ˙ (0) δ$ proves (eq:H)–(eq:Hab). Equation (eq:gaugedefect) is compact integration by parts.

Displayed formulas in this passage

V-B@1#eq:H · line 492
$H = γ + δ (t) a + δ^{'} (t) b_{0},$
V-B@1#eq:Hab · line 497
$γ^{ν} a^{ν} b_{0}^{ν} = f \partial^{ν} ϕ, = u ˙_{0} \partial^{ν} ϕ_{0} - u_{0} \partial_{t} \partial^{ν} ϕ ∣_{0}, = u_{0} \partial^{ν} ϕ_{0} .$
V-B@1#eq:gaugedefect · line 505
$B_{f} (K ξ, ϕ) = ⟨ H, ξ ⟩ .$
unlabeled · line 513
$2 \partial_{μ} (\partial^{(μ} v \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (v \partial^{α} ϕ)) = (P v) \partial^{ν} ϕ,$
unlabeled · line 522
$P (Θ u) = - f - δ (t) u ˙_{0} - δ^{'} (t) u_{0} .$

Theorem: Complete defect

Displayed formulas in this passage

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