Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

The completion Green operators and free algebra

Frozen analytical data and predecessor handoff › The completion Green operators and free algebra · lines 472–537

X-C-ST@1#eq:Green, X-C-ST@1#eq:free, X-C-ST@1#eq:separation

Write

D = C_{c}^{\infty} (M; C) .

It has its standard LF topology, with smooth seminorms on each fixed compact-support stage.

The positive free Green operators satisfy

P_{m} \ERc = P_{m} \EAc = c, E_{R} P_{m} a = E_{A} P_{m} a = a, E_{R}^{T} = E_{A}, supp \ERc \subset J_{η}^{+} (supp c), supp \EAc \subset J_{η}^{-} (supp c) .

They act continuously from compact smooth sources to smooth solutions. These are completion-space Green-operator statements BGP; no conical Cauchy theorem is used.

Retain X-B-ST's scalar free Wick algebra and quotient

A_{0} = A_{off} / J_{0},

where $J_{0}$ is generated by $F_{P_{m} a}$ , $a \in D$ . For

F_{c} (ψ) = \int c ψ d^{4} x, ϕ (c) = [F_{c}],

the degree-one relations are

ϕ (P_{m} a) = 0, [ϕ (c), ϕ (d)] = i ℏ ⟨ c, (E_{A} - E_{R}) d ⟩ 1 .

The degree-one topology is the quotient test topology on $D / P_{m} D$ .

The insertion domain is instead

L_{off} = {F_{c} : c \in D} ≅ D .

It consists of actual off-shell representatives. Output classes are not used as unspecified new input representatives.

The inherited degree-one separation property is

ϕ (d) = 0 ⟺ d \in P_{m} D ⟺ ⟨ d, u ⟩ = 0 for every smooth spatially compact free solution u .

This is X-B-ST's normalization standard XBST (Section C3); it is also the Green-operator exact-sequence mechanism BGP (Section 3.4).

Displayed formulas in this passage

X-C-ST@1#eq:Green · line 482
$P_{m} \ERc = P_{m} \EAc = c, E_{R} P_{m} a = E_{A} P_{m} a = a, E_{R}^{T} = E_{A}, supp \ERc \subset J_{η}^{+} (supp c), supp \EAc \subset J_{η}^{-} (supp c) .$
X-C-ST@1#eq:free · line 506
$ϕ (P_{m} a) = 0, [ϕ (c), ϕ (d)] = i ℏ ⟨ c, (E_{A} - E_{R}) d ⟩ 1 .$
X-C-ST@1#eq:separation · line 524
$ϕ (d) = 0 ⟺ d \in P_{m} D ⟺ ⟨ d, u ⟩ = 0 for every smooth spatially compact free solution u .$
unlabeled · line 475
$D = C_{c}^{\infty} (M; C) .$
unlabeled · line 496
$A_{0} = A_{off} / J_{0},$
unlabeled · line 501
$F_{c} (ψ) = \int c ψ d^{4} x, ϕ (c) = [F_{c}],$
unlabeled · line 516
$L_{off} = {F_{c} : c \in D} ≅ D .$