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

The smeared relational functional

One classical observable and its actual Cauchy data › The smeared relational functional · lines 401–429

VII-I@1#eq:observable-coefficients

For real $f \in C_{c}^{\infty} (M_{+}^{\circ})$ , define formally

O_{f} [q, ϕ] = \int f (y) ϕ (X_{q}^{- 1} (y)) d^{4} y .

The leading map is the identity, so formal inversion is unique. Its coefficients are

O_{f}^{(0)} O_{f}^{(1)} O_{f}^{(2)} = \int f ϕ, = B_{f} = - \int f w^{μ} [h] \partial_{μ} ϕ, = \int f [(w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ] .

All integrals in this paper use $d^{4} x$ unless indicated otherwise.

For $K = supp f$ , the metric dependence lies in

D_{f} = J^{-} (K) \cap {t \geq 0},

a compact set. Nested retarded operations do not enlarge this overall causal past.

Displayed formulas in this passage

VII-I@1#eq:observable-coefficients · line 410
$O_{f}^{(0)} O_{f}^{(1)} O_{f}^{(2)} = \int f ϕ, = B_{f} = - \int f w^{μ} [h] \partial_{μ} ϕ, = \int f [(w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ] .$
unlabeled · line 404
$O_{f} [q, ϕ] = \int f (y) ϕ (X_{q}^{- 1} (y)) d^{4} y .$
unlabeled · line 424
$D_{f} = J^{-} (K) \cap {t \geq 0},$