Manuscript XII-C@1 · XII-C.tex · section · dead1004

The copied pure-trace check

C1: the density representative and actual cubic source › The copied pure-trace check · lines 768–795

XII-C@1#eq:full-trace-sector

XII-B's identity (eq:input-trace) is a consistency requirement, not the general source formula. For $γ = ψ η$ , algebraic substitution gives $C_{1} = C_{2} = 0$ and $Γ_{2} = - ψ Γ_{1}$ . Consequently (eq:actualsource) gives

D_{σ}_{ε}^{μν}_{γ = ψ η} = - ψ [η^{μν} (\partial_{ρ} σ_{ε}) \partial^{ρ} ψ + (\partial^{μ} σ_{ε}) \partial^{ν} ψ + (\partial^{ν} σ_{ε}) \partial^{μ} ψ] .

Contracting with $η_{μν}$ reproduces (eq:input-trace). Thus the predecessor's nonzero layer term is retained.

Density representatives cannot be changed silently. If $ℓ_{3}$ is replaced by $ℓ_{3} + \partial_{μ} B^{μ}$ , the weighted source changes by $- EL_{γ} ((\partial_{μ} σ_{ε}) B^{μ})$ , as proved in XII-B. Equation (eq:ell3) eliminates that ambiguity.

Displayed formulas in this passage

XII-C@1#eq:full-trace-sector · line 775
$D_{σ}_{ε}^{μν}_{γ = ψ η} = - ψ [η^{μν} (\partial_{ρ} σ_{ε}) \partial^{ρ} ψ + (\partial^{μ} σ_{ε}) \partial^{ν} ψ + (\partial^{ν} σ_{ε}) \partial^{μ} ψ] .$