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

Off-shell functional

Relational expansion and retained infinitesimal defects › Off-shell functional · lines 1158–1192

X-A-conv@1#eq:inverse, X-A-conv@1#eq:observable, X-A-conv@1#eq:Jw

The formal inverse of (eq:Xexp) is

Y^{μ} (z) = z^{μ} - κ w^{μ} (z) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (z) (mod κ^{3}) .

It is obtained by substitution in $X (Y (z)) = z$ ; no numerical global inverse is required.

Define

Φ_{g, ε} [h, k, ϕ] (z) = ϕ (Y (z)), O_{f, g, ε} [h, k, ϕ] = \int f (z) Φ_{g, ε} (z) d^{4} z .

The test $f$ is fixed in relational-label space. It is not transformed as a spacetime-coordinate test.

For a vector $w$ , write

J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .

Displayed formulas in this passage

X-A-conv@1#eq:inverse · line 1161
$Y^{μ} (z) = z^{μ} - κ w^{μ} (z) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (z) (mod κ^{3}) .$
X-A-conv@1#eq:observable · line 1173
$Φ_{g, ε} [h, k, ϕ] (z) = ϕ (Y (z)), O_{f, g, ε} [h, k, ϕ] = \int f (z) Φ_{g, ε} (z) d^{4} z .$
X-A-conv@1#eq:Jw · line 1185
$J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .$