Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Complete Ward defect

Raw current and complete gauge defect › The defect covector · lines 765–818 · proof included or linked

V@1#thm:defect, V@1#eq:divraw, V@1#eq:H, V@1#eq:Hab

Distributionally,

\partial_{μ} J_{raw}^{μν} = \frac{1}{2} (P V) \partial^{ν} ϕ, P V = - f - δ (t) u ˙_{0} - δ^{'} (t) u_{0} .

Consequently

H := K^{*} J_{raw} = γ + δ (t) a + δ^{'} (t) b,

where the covector-valued coefficients are

γ_{ν} a_{ν} b_{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} .

The density $γ$ is smooth and compact in $0 < t < T$ ; $a, b$ are smooth compact spatial data.

Proof. For any scalar distribution $w$ , the product rule gives

2 \partial_{μ} [\partial^{(μ} w \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (w \partial^{α} ϕ)] = (□ w) \partial^{ν} ϕ - w \partial^{ν} □ ϕ .

Use $□ ϕ = m^{2} ϕ$ . Also

\partial_{t}^{2} (Θ u) = Θ u ¨ + δ (t) u ˙_{0} + δ^{'} (t) u_{0} .

This proves (eq:divraw). Apply (eq:Kstar), then use

c (t) δ^{'} (t) = c (0) δ^{'} (t) - c ˙ (0) δ (t) .

The resulting coefficients are (eq:Hab).

Displayed formulas in this passage

V@1#eq:divraw · line 768
$\partial_{μ} J_{raw}^{μν} = \frac{1}{2} (P V) \partial^{ν} ϕ, P V = - f - δ (t) u ˙_{0} - δ^{'} (t) u_{0} .$
V@1#eq:H · line 775
$H := K^{*} J_{raw} = γ + δ (t) a + δ^{'} (t) b,$
V@1#eq:Hab · line 783
$γ_{ν} a_{ν} b_{ν} = f \partial_{ν} ϕ, = u ˙_{0} \partial_{ν} ϕ ∣_{0} - u_{0} \partial_{t} \partial_{ν} ϕ ∣_{0}, = u_{0} \partial_{ν} ϕ ∣_{0} .$
unlabeled · line 798
$2 \partial_{μ} [\partial^{(μ} w \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (w \partial^{α} ϕ)] = (□ w) \partial^{ν} ϕ - w \partial^{ν} □ ϕ .$
unlabeled · line 808
$\partial_{t}^{2} (Θ u) = Θ u ¨ + δ (t) u ˙_{0} + δ^{'} (t) u_{0} .$
unlabeled · line 814
$c (t) δ^{'} (t) = c (0) δ^{'} (t) - c ˙ (0) δ (t) .$