Manuscript XI-D@1 · XI-D.tex · section · 994a5c1b

Classical scalar completion

The inherited held-fixed-metric defect › Classical scalar completion · lines 885–937

XI-D@1#eq:A1, XI-D@1#eq:A2

For a smooth tensor $r$ put

b_{1} [r] = \frac{1}{2} tr_{η} r, V_{1, g} [r] u = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} u) + g m^{2} b_{1} [r] u .

For a free scalar $ϕ$ define

A_{1} [r; ϕ] = \int f (- E_{m}^{R} V_{1, g} [r] ϕ - X_{1} [r] \cdot \partial ϕ) d^{4} x .

This is the scalar-only first coefficient of X-A.

Let $H$ and $k$ be the first and second metric coefficients. Use X-A's density coefficient $a_{2} [H]$ and define

U_{ε} [H] u = \partial_{α} (σ_{ε} a_{1}^{α β} [H] \partial_{β} u),

X_{2}^{μ} [H, k] = - E_{0}^{R} (U_{ε} [H] X_{1}^{μ} [H] + \partial_{α} (σ_{ε} (a_{1}^{α μ} [k] + a_{2}^{α μ} [H]))) .

Let $V_{2, g} [H, k]$ be X-A's switched second scalar operator and set

ρ_{1} = - E_{m}^{R} V_{1, g} [H] ϕ, ρ_{2} = E_{m}^{R} V_{1, g} [H] E_{m}^{R} V_{1, g} [H] ϕ - E_{m}^{R} V_{2, g} [H, k] ϕ .

The completed classical coefficient is

A_{2}^{sc} [H, k; ϕ] = \int f [ρ_{2} - X_{1} [H] \cdot \partial ρ_{1} + (X_{1} [H] \cdot \partial X_{1} [H] - X_{2} [H, k]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [H] X_{1}^{ν} [H] \partial_{μ} \partial_{ν} ϕ] d^{4} x .

All these expressions are classical evaluations on actual smooth configurations.

Displayed formulas in this passage

XI-D@1#eq:A1 · line 895
$A_{1} [r; ϕ] = \int f (- E_{m}^{R} V_{1, g} [r] ϕ - X_{1} [r] \cdot \partial ϕ) d^{4} x .$
XI-D@1#eq:A2 · line 923
$A_{2}^{sc} [H, k; ϕ] = \int f [ρ_{2} - X_{1} [H] \cdot \partial ρ_{1} + (X_{1} [H] \cdot \partial X_{1} [H] - X_{2} [H, k]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [H] X_{1}^{ν} [H] \partial_{μ} \partial_{ν} ϕ] d^{4} x .$
unlabeled · line 888
$b_{1} [r] = \frac{1}{2} tr_{η} r, V_{1, g} [r] u = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} u) + g m^{2} b_{1} [r] u .$
unlabeled · line 905
$U_{ε} [H] u = \partial_{α} (σ_{ε} a_{1}^{α β} [H] \partial_{β} u),$
unlabeled · line 909
$X_{2}^{μ} [H, k] = - E_{0}^{R} (U_{ε} [H] X_{1}^{μ} [H] + \partial_{α} (σ_{ε} (a_{1}^{α μ} [k] + a_{2}^{α μ} [H]))) .$
unlabeled · line 917
$ρ_{1} = - E_{m}^{R} V_{1, g} [H] ϕ, ρ_{2} = E_{m}^{R} V_{1, g} [H] E_{m}^{R} V_{1, g} [H] ϕ - E_{m}^{R} V_{2, g} [H, k] ϕ .$