Manuscript III@2 · III.tex · theorem · 82dad3e5

Theorem: Descent on the coupled jet

The coupled first-order jet and retarded harmonic section › The complete quotient functional · lines 1335–1364 · proof included or linked

III@2#thm:jet-descent

The functional $F_{f}$ descends to $J / G_{R}$ . If

(k, ϕ_{0}, ψ_{1}) = S_{R} (h, ϕ_{0}, ϕ_{1}),

then

F_{f} (h, ϕ_{0}, ϕ_{1}) = \int_{M} f (ϕ_{0} + κ ψ_{1}) d^{4} x, P ψ_{1} = - L_{k} ϕ_{0} .

Proof. Both coefficients are invariant by the section calculation, so the functional factors through the orbit map. On the slice, $w_{R} (k) = 0$ .

Writing $w = w_{R} (h)$ ,

P (ϕ_{1} - X_{w} ϕ_{0}) + L_{h - K w} ϕ_{0} = P ϕ_{1} - P X_{w} ϕ_{0} + L_{h} ϕ_{0} + [P, X_{w}] ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} - X_{w} P ϕ_{0} = 0.

Displayed formulas in this passage

unlabeled · line 1339
$(k, ϕ_{0}, ψ_{1}) = S_{R} (h, ϕ_{0}, ϕ_{1}),$
unlabeled · line 1343
$F_{f} (h, ϕ_{0}, ϕ_{1}) = \int_{M} f (ϕ_{0} + κ ψ_{1}) d^{4} x, P ψ_{1} = - L_{k} ϕ_{0} .$
unlabeled · line 1357
$P (ϕ_{1} - X_{w} ϕ_{0}) + L_{h - K w} ϕ_{0} = P ϕ_{1} - P X_{w} ϕ_{0} + L_{h} ϕ_{0} + [P, X_{w}] ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} - X_{w} P ϕ_{0} = 0.$