Manuscript VII-I@1 · VII-I.tex · theorem · 907e96ab

Theorem: Linear sharp coordinates in the free quantum algebra

Route B3: constrained auxiliary coordinates › A genuine linear free-shell quantum repair · lines 1701–1752 · proof included or linked

VII-I@1#thm:linear-quantum

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The tensor $q_{sm, a}$ is smooth and compact, depends linearly on $a$ , and satisfies

q_{a} - q_{sm, a} = □ r_{a}

for a compact distribution $r_{a}$ . Therefore

ℓ_{a} [h] = h (q_{sm, a}) for every smooth □ h = 0.

The assignment

ℓ_{a} = h (q_{sm, a})

is a well-defined linear element of the ordinary free on-shell field algebra. Finite products, adjoints, and their free contractions exist. Its class is independent of the chosen time cutoff.

Proof. Smoothness follows from Lemma lem:linear-smoothing. The commutator with a time cutoff is supported in a bounded time band. Spacelike compactness makes its support compact.

The retarded and advanced identities are

E^{R}_{0} q_{sm, a} = ρ_{0} u_{a}, E^{A}_{0} q_{sm, a} = (ρ_{0} - 1) u_{a} .

Hence $E_{0} (q_{a} - q_{sm, a}) = 0$ . The retarded and advanced solutions of that compact source agree, so

r_{a} = E^{R}_{0} (q_{a} - q_{sm, a}) = E^{A}_{0} (q_{a} - q_{sm, a})

has compact support and $□ r_{a} = q_{a} - q_{sm, a}$ . Pair with a smooth homogeneous field to obtain the classical shell equality.

The smooth representative is an ordinary compact field test, so the usual free on-shell algebra admits it. Two choices have a smooth compact difference $d$ with $E_{0} d = 0$ . Then $d = □ (E^{R}_{0} d)$ , with $E^{R}_{0} d$ smooth compact. The free equation ideal and the bisolution contraction kill that difference.

Displayed formulas in this passage

unlabeled · line 1705
$q_{a} - q_{sm, a} = □ r_{a}$
unlabeled · line 1709
$ℓ_{a} [h] = h (q_{sm, a}) for every smooth □ h = 0.$
unlabeled · line 1715
$ℓ_{a} = h (q_{sm, a})$
unlabeled · line 1731
$E^{R}_{0} q_{sm, a} = ρ_{0} u_{a}, E^{A}_{0} q_{sm, a} = (ρ_{0} - 1) u_{a} .$
unlabeled · line 1738
$r_{a} = E^{R}_{0} (q_{a} - q_{sm, a}) = E^{A}_{0} (q_{a} - q_{sm, a})$

Theorem: Linear sharp coordinates in the free quantum algebra

Displayed formulas in this passage

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