Manuscript IX-D-ST@1 · IX-D-ST.tex · proposition · b1dc631a

Proposition: Locked first-stage commutator

Domain, ordering geometry, and the commutator · lines 1162–1185 · proof included or linked

IX-D-ST@1#prop:commutator, IX-D-ST@1#eq:commutator

On the first-stage sector,

B_{f} ⋆_{c} W_{g} - W_{g} ⋆_{c} B_{f} C_{f, g} = ℏ C_{f, g}, = - 2 ⟨ f g (K - K), \partial_{t} ϕ (x) h_{00} (y) ⟩ = - 4 i ⟨ f g Im K, \partial_{t} ϕ (x) h_{00} (y) ⟩ .

Proof. Subtract (eq:firststages). Their classical terms coincide. Only smooth configuration factors are reordered. Distributional linearity and $K - K = 2 i Im K$ give the result.

Displayed formulas in this passage

IX-D-ST@1#eq:commutator · line 1165
$B_{f} ⋆_{c} W_{g} - W_{g} ⋆_{c} B_{f} C_{f, g} = ℏ C_{f, g}, = - 2 ⟨ f g (K - K), \partial_{t} ϕ (x) h_{00} (y) ⟩ = - 4 i ⟨ f g Im K, \partial_{t} ϕ (x) h_{00} (y) ⟩ .$

Nearby in this section

section: Domain, ordering geometry, and the commutator
proposition: Proposition: Domain stability
section: Domain, ordering geometry, and the commutator
proposition: Proposition: Double ordering
section: Domain, ordering geometry, and the commutator
lemma: Lemma: A strongly separated comparison class
section: Domain, ordering geometry, and the commutator
section: Domain, ordering geometry, and the commutator