Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

One fixed compact-test space

Frozen data, completed tests, and the topology sheet › One fixed compact-test space · lines 521–557

X-D-ST@1#eq:C, X-D-ST@1#eq:quotient

Let

C = K \cup (supp g \cap J_{η}^{-} (K)) .

This compact set is independent of the ramp width and lies above $t = 2 δ$ .

For a compact $C \subset M$ , let

D_{C} = {d \in C_{c}^{\infty} (M; C) : supp d \subset C}, p_{N} (d) = ∣ α ∣ \leq N max M sup ∣ \partial^{α} d ∣.

The derivatives are ambient smooth derivatives. No smooth structure is assigned to the possibly nonsmooth boundary of $C$ .

The inherited degree-one free-field sector has topology

Q_{m} = D / P_{m} D, D = C_{c}^{\infty} (M; C), ϕ (d) = [F_{d}] .

It is the linear sector of the scalar free algebra $A_{0}$ . Its equation relation is $ϕ (P_{m} a) = 0$ , and its commutator is

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

No topology on a newly enlarged interacting algebra is introduced FR.

Displayed formulas in this passage

X-D-ST@1#eq:C · line 524
$C = K \cup (supp g \cap J_{η}^{-} (K)) .$
X-D-ST@1#eq:quotient · line 542
$Q_{m} = D / P_{m} D, D = C_{c}^{\infty} (M; C), ϕ (d) = [F_{d}] .$
unlabeled · line 532
$D_{C} = {d \in C_{c}^{\infty} (M; C) : supp d \subset C}, p_{N} (d) = ∣ α ∣ \leq N max M sup ∣ \partial^{α} d ∣.$
unlabeled · line 551
$[ϕ (c), ϕ (d)] = i ℏ ⟨ c, (E_{A} - E_{R}) d ⟩ 1 .$