Manuscript XII-D@1 · XII-D.tex · section · 308a2080

D2: both completion legs and the compact root

D2: both completion legs and the compact root · lines 594–637

XII-D@1#eq:completion-legs, XII-D@1#eq:Aone

The completion is written explicitly so that no unspecified predecessor formula is needed. Set

a^{μν, ab} = \frac{1}{2} η^{μν} η^{ab} - η^{μ (a} η^{b) ν}, a_{1}^{μν} [r] = a^{μν, ab} r_{ab}, b_{1} [r] = \frac{1}{2} η^{ab} r_{ab} .

The divergence-form coordinate coefficient and the scalar equation coefficient are

X_{1}^{μ} [r] V_{1, g} [r] ϕ ϕ_{1} [r; ϕ] = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} (V_{1, g} [r] ϕ) .

They follow from the inherited coordinate prescription and the minimally coupled switched scalar action. In particular, derivatives of $g$ are retained inside the displayed divergence.

For $f \in C_{c}^{\infty} (M_{T}^{\circ})$ define

A_{1} [f, g] (r; ϕ) = \int f (x) (ϕ_{1} [r; ϕ] (x) - X_{1}^{μ} [r] (x) \partial_{μ} ϕ (x)) d^{4} x .

The scalar field is arbitrary and smooth at this stage.

Displayed formulas in this passage

XII-D@1#eq:completion-legs · line 612
$X_{1}^{μ} [r] V_{1, g} [r] ϕ ϕ_{1} [r; ϕ] = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} (V_{1, g} [r] ϕ) .$
XII-D@1#eq:Aone · line 628
$A_{1} [f, g] (r; ϕ) = \int f (x) (ϕ_{1} [r; ϕ] (x) - X_{1}^{μ} [r] (x) \partial_{μ} ϕ (x)) d^{4} x .$
unlabeled · line 600
$a^{μν, ab} = \frac{1}{2} η^{μν} η^{ab} - η^{μ (a} η^{b) ν}, a_{1}^{μν} [r] = a^{μν, ab} r_{ab}, b_{1} [r] = \frac{1}{2} η^{ab} r_{ab} .$

Nearby in this section

definition: Definition: Completed new insertion