Manuscript XI-B@1 · XI-B.tex · section · 063b989e

Operators copied from XI-A

The XI-A handoff and convention sheet › Operators copied from XI-A · lines 360–413

XI-B@1#eq:basic, XI-B@1#eq:handoff, XI-B@1#eq:vertex

For a symmetric tensor $γ$ define

γ ˉ_{μν} = γ_{μν} - \frac{1}{2} η_{μν} tr_{η} γ, D^{ν} γ = \partial_{μ} γ ˉ^{μν},

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .

Let $P$ denote trace reversal, so $γ ˉ = P γ$ . In four dimensions,

P^{2} = id, D (K ξ) = □ ξ, G_{lin} (K ξ) = 0.

XI-A gives

2 G_{lin} (γ) + E_{gf} (γ) = - □ P γ, S_{gf} = \int_{M_{T}} D_{ν} γ D^{ν} γ d^{4} x .

This is not the normalization $- \frac{1}{2} □ P$ .

The inherited interaction is

V_{1, g} [γ, ϕ] = - \frac{1}{2} \int_{M_{T}} g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x,

with

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{ρ} ϕ \partial^{ρ} ϕ - m^{2} ϕ^{2}) .

It is not inserted into the free action or into the free differential.

The invariant XI-A action has a fixed-background Noether identity; its gauge-fixed action has an additional breaking term. The switching contribution includes $(\partial_{μ} g) T_{0}^{μν}$ . None of these facts is replaced by free gauge-fixed solvability XIA.

Displayed formulas in this passage

XI-B@1#eq:basic · line 375
$P^{2} = id, D (K ξ) = □ ξ, G_{lin} (K ξ) = 0.$
XI-B@1#eq:handoff · line 383
$2 G_{lin} (γ) + E_{gf} (γ) = - □ P γ, S_{gf} = \int_{M_{T}} D_{ν} γ D^{ν} γ d^{4} x .$
XI-B@1#eq:vertex · line 393
$V_{1, g} [γ, ϕ] = - \frac{1}{2} \int_{M_{T}} g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x,$
unlabeled · line 363
$γ ˉ_{μν} = γ_{μν} - \frac{1}{2} η_{μν} tr_{η} γ, D^{ν} γ = \partial_{μ} γ ˉ^{μν},$
unlabeled · line 369
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .$
unlabeled · line 400
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{ρ} ϕ \partial^{ρ} ϕ - m^{2} ϕ^{2}) .$