Manuscript VII-I@1 · VII-I.tex · proposition · 907e96ab

Proposition: Classical presentation identity

One classical observable and its actual Cauchy data › The smeared relational functional · lines 430–464 · proof included or linked

VII-I@1#prop:presentations, VII-I@1#eq:jacobian-presentation

Back to manuscript

Through order $κ^{2}$ ,

O_{f} [q, ϕ] = \int det (D X_{q} (x)) f (X_{q} (x)) ϕ (x) d^{4} x .

The determinant has leading term $1$ ; no absolute-value ambiguity arises in the formal orientation-preserving expansion.

Proof. Formal change of variables gives the identity. Explicitly,

det D X_{q} = 1 + κ \partial_{μ} w^{μ} + κ^{2} [\partial_{μ} v^{μ} + \frac{1}{2} ((\partial_{μ} w^{μ})^{2} - \partial_{μ} w^{ν} \partial_{ν} w^{μ})] .

Multiply this by the Taylor expansion of $f (x + κ w + κ^{2} v)$ . At order one, integration by parts gives $- \int f w^{μ} \partial_{μ} ϕ$ . At order two, two integrations by parts give precisely (eq:observable-coefficients).

The observation factor has compact support strictly above $Σ$ . These integrations occur in the observation variable. They do not delete the initial-surface terms obtained when a coordinate kernel is subsequently transposed in a metric variable.

Displayed formulas in this passage

VII-I@1#eq:jacobian-presentation · line 433
$O_{f} [q, ϕ] = \int det (D X_{q} (x)) f (X_{q} (x)) ϕ (x) d^{4} x .$
unlabeled · line 445
$det D X_{q} = 1 + κ \partial_{μ} w^{μ} + κ^{2} [\partial_{μ} v^{μ} + \frac{1}{2} ((\partial_{μ} w^{μ})^{2} - \partial_{μ} w^{ν} \partial_{ν} w^{μ})] .$

Nearby in this section

section: The smeared relational functional