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

The classical solution spaces

Locked signs, solution spaces, and Green operators › The classical solution spaces · lines 551–572

V@1#eq:gaugeannihilation, V@1#eq:Kstar, V@1#eq:Dstar

For every smooth $ξ$ and every $J \in E$ ,

F_{J} (K ξ) = - 2 \int ξ_{ν} \partial_{μ} J^{μν} d^{4} x = 0.

This elementary identity is the necessary condition used in every negative result.

The formal adjoints, with covectors paired against vectors, are

(K^{*} J)_{ν} (D^{*} a)^{μν} = - 2 η_{ν β} \partial_{μ} J^{μ β}, = - \partial^{(μ} a^{ν)} + \frac{1}{2} η^{μν} \partial_{α} a^{α}, K^{*} D^{*} a = □ a .

Displayed formulas in this passage

V@1#eq:gaugeannihilation · line 554
$F_{J} (K ξ) = - 2 \int ξ_{ν} \partial_{μ} J^{μν} d^{4} x = 0.$
V@1#eq:Kstar · line 563
$(K^{*} J)_{ν} (D^{*} a)^{μν} = - 2 η_{ν β} \partial_{μ} J^{μ β}, = - \partial^{(μ} a^{ν)} + \frac{1}{2} η^{μν} \partial_{α} a^{α}, K^{*} D^{*} a = □ a .$
V@1#eq:Dstar · line 563
$(K^{*} J)_{ν} (D^{*} a)^{μν} = - 2 η_{ν β} \partial_{μ} J^{μ β}, = - \partial^{(μ} a^{ν)} + \frac{1}{2} η^{μν} \partial_{α} a^{α}, K^{*} D^{*} a = □ a .$