Manuscript Em03-A@1 · Em03-A-v1.tex · proposition · 6d54627d

Proposition: Finite-sector underdetermination

Why the supplied landmarks do not substitute for this proof › Energy-preserving finite isometries do not select a spectrum · lines 1473–1512 · proof included or linked

Em03-A@1#prop:finite

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Energy preservation and the isometry condition alone do not select a Hawking occupation law or a dimensional emission rate.

Proof. Fix a source energy $E$ , a source degeneracy label $a = 1, \dots, d$ , and permitted radiation energies $ω_{1}, \dots, ω_{k} \leq E$ . Choose orthonormal daughter–radiation states

∣ E - ω_{j}, a ⟩_{D} \otimes ∣ ω_{j}, j ⟩_{R}

whose total energy is $E$ . The additional label $j$ may distinguish degenerate radiation sectors.

For every probability vector $w = (w_{1}, \dots, w_{k})$ , define

V_{w} ∣ E, a ⟩ = j = 1 \sum^{k} w_{j} ∣ E - ω_{j}, a ⟩_{D} \otimes ∣ ω_{j}, j ⟩_{R} .

Orthogonality gives

V_{w}^{*} V_{w} = I, (H_{D} + H_{R}) V_{w} = V_{w} H_{S} .

The radiation-energy probabilities are exactly $w_{j}$ .

For $k \geq 2$ , two different probability vectors give different spectra while preserving both displayed identities. Nothing in these identities singles out a Bose factor or a transmission probability. Moreover, no duration or evolution schedule has been specified, so they cannot determine a rate per unit time.

Displayed formulas in this passage

unlabeled · line 1484
$∣ E - ω_{j}, a ⟩_{D} \otimes ∣ ω_{j}, j ⟩_{R}$
unlabeled · line 1492
$V_{w} ∣ E, a ⟩ = j = 1 \sum^{k} w_{j} ∣ E - ω_{j}, a ⟩_{D} \otimes ∣ ω_{j}, j ⟩_{R} .$
unlabeled · line 1500
$V_{w}^{*} V_{w} = I, (H_{D} + H_{R}) V_{w} = V_{w} H_{S} .$

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section: Energy-preserving finite isometries do not select a spectrum