Manuscript XI-F@2 · XI-F-v2.tex · section · f4899be0

Coordinate operator, first and second coefficients

Copied sheet › Coordinate operator, first and second coefficients · lines 452–497

XI-F@2#eq:X1, XI-F@2#eq:affine, XI-F@2#eq:XAv

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r, C^{μ} [r] = - \partial_{α} a_{1}^{α μ} [r] = D^{μ} r,

X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .

V_{1, g} [r] u = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} u) + g m^{2} b_{1} [r] u, A_{1} [r; ϕ] = \int_{M_{T}} f (- E_{m}^{R} V_{1, g} [r] ϕ - X_{1} [r] \cdot \partial ϕ) d^{4} x .

$A_{2}^{sc} [H, k; ϕ]$ is X-A's complete on-shell second coefficient, written out in XI-D: with $ρ_{1} = - E_{m}^{R} V_{1, g} [H] ϕ$ , $ρ_{2} = E_{m}^{R} V_{1, g} [H] E_{m}^{R} V_{1, g} [H] ϕ - E_{m}^{R} V_{2, g} [H, k] ϕ$ ,

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 .

Two copied facts about it:

A_{2}^{sc} [H, k + r; ϕ] - A_{2}^{sc} [H, k; ϕ] A_{1} [h^{'}; ϕ_{0}] = A_{1} [h; ϕ_{0}], A_{2}^{sc} [h^{'}, k^{'}; ϕ_{0}] = A_{1} [r; ϕ] (XI-D, affine lemma), = A_{2}^{sc} [h, k; ϕ_{0}] (X-A, Theorem (v)),

for every free $ϕ_{0}$ , every smooth $(h, k)$ , and

h^{'} = h + K ξ, k^{'} = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ} K ξ, ξ, ζ \in g_{U} .

The free input $ϕ_{0}$ is not transformed in (eq:XAv); it is held fixed, exactly as XI-E held the scalar fixed in its cubic variation.

Displayed formulas in this passage

XI-F@2#eq:X1 · line 459
$X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .$
XI-F@2#eq:affine · line 479
$A_{2}^{sc} [H, k + r; ϕ] - A_{2}^{sc} [H, k; ϕ] A_{1} [h^{'}; ϕ_{0}] = A_{1} [h; ϕ_{0}], A_{2}^{sc} [h^{'}, k^{'}; ϕ_{0}] = A_{1} [r; ϕ] (XI-D, affine lemma), = A_{2}^{sc} [h, k; ϕ_{0}] (X-A, Theorem (v)),$
XI-F@2#eq:XAv · line 479
$A_{2}^{sc} [H, k + r; ϕ] - A_{2}^{sc} [H, k; ϕ] A_{1} [h^{'}; ϕ_{0}] = A_{1} [h; ϕ_{0}], A_{2}^{sc} [h^{'}, k^{'}; ϕ_{0}] = A_{1} [r; ϕ] (XI-D, affine lemma), = A_{2}^{sc} [h, k; ϕ_{0}] (X-A, Theorem (v)),$
unlabeled · line 454
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r, C^{μ} [r] = - \partial_{α} a_{1}^{α μ} [r] = D^{μ} r,$
unlabeled · line 464
$V_{1, g} [r] u = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} u) + g m^{2} b_{1} [r] u, A_{1} [r; ϕ] = \int_{M_{T}} f (- E_{m}^{R} V_{1, g} [r] ϕ - X_{1} [r] \cdot \partial ϕ) d^{4} x .$
unlabeled · line 472
$A_{2}^{sc} [H, k; ϕ] = \int f [ρ_{2} - X_{1} [H] \cdot \partial ρ_{1} + (X_{1} [H] \cdot \partial X_{1} [H] - X_{2} [H, k]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [H] X_{1}^{ν} [H] \partial_{μ} \partial_{ν} ϕ] d^{4} x .$
unlabeled · line 490
$h^{'} = h + K ξ, k^{'} = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ} K ξ, ξ, ζ \in g_{U} .$