Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Formal inverse and observable

Relational expansion and covariance: Route I › Formal inverse and observable · lines 996–1033

X-A-ST@1#eq:inverse, X-A-ST@1#eq:observable, X-A-ST@1#eq:relationalcoeff

The coordinate inverse exists algebraically:

Co_{q}^{- 1} (z) = z - κ w (z) + κ^{2} (w^{ν} \partial_{ν} w - v) (z) .

No globally invertible numerical coordinate map is required.

Define

Φ_{g, ε} [q, ϕ] (z) = ϕ (Co_{q}^{- 1} (z)), O_{f, g, ε} [q, ϕ] = \int f (z) Φ_{g, ε} [q, ϕ] (z) d^{4} z .

The label-space test $f$ is fixed, not pulled back. No $q$ -volume density is inserted in this pairing. The off-shell expression depends on $g$ only through the declared gauge and evaluation contract.

Taylor expansion gives

Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

For the dynamical evaluation, replace $ϕ_{1}, ϕ_{2}$ by $ρ_{1, g}, ρ_{2, g}$ . In particular, the forced scalar correction is not omitted.

Displayed formulas in this passage

X-A-ST@1#eq:inverse · line 999
$Co_{q}^{- 1} (z) = z - κ w (z) + κ^{2} (w^{ν} \partial_{ν} w - v) (z) .$
X-A-ST@1#eq:observable · line 1008
$Φ_{g, ε} [q, ϕ] (z) = ϕ (Co_{q}^{- 1} (z)), O_{f, g, ε} [q, ϕ] = \int f (z) Φ_{g, ε} [q, ϕ] (z) d^{4} z .$
X-A-ST@1#eq:relationalcoeff · line 1021
$Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$