Manuscript X-C@3 · X-C-v3.tex · theorem · 2260f5f8

Theorem: Existence, support, continuity, and normalization

C2 and C3: maps, transposition, signs, and polarization · lines 714–763 · proof included or linked

X-C@3#thm:maps, X-C@3#eq:normalization

The two maps (eq:second-map)–(eq:iteration) have smooth compact output tests supported in

supp g \cap J^{-} (supp c) .

They are continuous in compact-test and smooth background topologies. For every smooth $u$ , and hence for every smooth free $u$ ,

⟨ L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} c - L_{2, g} E_{m}^{A} c, u ⟩ = ⟨ c, E_{m}^{R} L_{1, g} E_{m}^{R} L_{1, g} u - E_{m}^{R} L_{2, g} u ⟩ .

Their sum on $F_{f}$ is therefore $\int f ρ_{2, g}$ , understood as the free field with the displayed compact test. No diagonal extension or subtraction is used.

Proof. The Green operators act continuously on compact smooth tests, with their respective causal supports; transposition exchanges the retarded and advanced operators BGP (Chapter 3). Consequently $E_{m}^{A} c$ is smooth, and applying $L_{2, g}$ produces a compact test supported in $supp g \cap J^{-} (supp c)$ .

The intermediate support $K_{1} = supp (L_{1, g} E_{m}^{A} c)$ is contained in the same set. The iterated output is supported in

supp g \cap J^{-} (K_{1}) \subset supp g \cap J^{-} (supp c) .

Continuous differentiation and multiplication give continuity. For input tests in a fixed compact support, every output lies in a fixed compact subset of $supp g$ .

Formal self-adjointness and transposition give

⟨ c, E_{m}^{R} L_{1, g} E_{m}^{R} L_{1, g} u ⟩ ⟨ c, E_{m}^{R} L_{2, g} u ⟩ = ⟨ L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} c, u ⟩, = ⟨ L_{2, g} E_{m}^{A} c, u ⟩ .

All source factors are compact because of $g$ . Every Green operation is applied to a smooth source or test; there is no singular product requiring extension.

Displayed formulas in this passage

X-C@3#eq:normalization · line 724
$⟨ L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} c - L_{2, g} E_{m}^{A} c, u ⟩ = ⟨ c, E_{m}^{R} L_{1, g} E_{m}^{R} L_{1, g} u - E_{m}^{R} L_{2, g} u ⟩ .$
unlabeled · line 754
$⟨ c, E_{m}^{R} L_{1, g} E_{m}^{R} L_{1, g} u ⟩ ⟨ c, E_{m}^{R} L_{2, g} u ⟩ = ⟨ L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} c, u ⟩, = ⟨ L_{2, g} E_{m}^{A} c, u ⟩ .$
unlabeled · line 718
$supp g \cap J^{-} (supp c) .$
unlabeled · line 745
$supp g \cap J^{-} (K_{1}) \subset supp g \cap J^{-} (supp c) .$

Theorem: Existence, support, continuity, and normalization

Displayed formulas in this passage

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