Manuscript X-A-conv@1 · X-A-conv.tex · proposition · c76fffa3

Proposition: Relational coefficients

Relational expansion and retained infinitesimal defects › Off-shell functional · lines 1193–1214 · proof included or linked

X-A-conv@1#prop:Ocoeff, X-A-conv@1#eq:Ozero, X-A-conv@1#eq:Oone, X-A-conv@1#eq:Otwo

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The coefficients of (eq:observable) are

O^{(0)} O^{(1)} O^{(2)} = \int f ϕ, = - \int f X_{w} ϕ, = \int f (J (w) ϕ - X_{v} ϕ) .

For the bare single-field functional, set $k = 0$ in these formulas.

Proof. Taylor-expand $ϕ (Y (z))$ using (eq:inverse). The linear displacement gives $- X_{w} ϕ$ . The second displacement gives $(w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ$ , and the quadratic Taylor term gives $\frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ$ .

Displayed formulas in this passage

X-A-conv@1#eq:Ozero · line 1196
$O^{(0)} O^{(1)} O^{(2)} = \int f ϕ, = - \int f X_{w} ϕ, = \int f (J (w) ϕ - X_{v} ϕ) .$
X-A-conv@1#eq:Oone · line 1196
$O^{(0)} O^{(1)} O^{(2)} = \int f ϕ, = - \int f X_{w} ϕ, = \int f (J (w) ϕ - X_{v} ϕ) .$
X-A-conv@1#eq:Otwo · line 1196
$O^{(0)} O^{(1)} O^{(2)} = \int f ϕ, = - \int f X_{w} ϕ, = \int f (J (w) ϕ - X_{v} ϕ) .$

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section: Off-shell functional