Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Endpoint completion and the no-remnant condition

Endpoint completion and the no-remnant condition · lines 1126–1158

Hawk01-B@1#eq:endpointenergy, Hawk01-B@1#eq:endpointsuccess, Hawk01-B@1#eq:endpointfailure

Choose a fixed final residual energy $m_{*}$ satisfying

0 \leq m_{*} < M - e_{K} - e .

The information-free endpoint residual is one fixed state $∣ v ⟩$ of energy $m_{*}$ . For completed residual basis states, introduce endpoint radiation states $∣ j, r ⟩_{*}$ of energy

E_{*}^{s} = M - e_{K} - e - m_{*} .

For pending residual basis states, introduce $∣ f, r, i ⟩_{*}$ of energy

E_{*}^{f} = M - e_{K} - m_{*} .

All these radiation states are orthogonal, with the specified conserved labels.

Define

V_{*} ∣ j, r ⟩_{B_{N}} V_{*} ∣ f, r; ψ ⟩_{B_{N}} = ∣ v ⟩_{B_{*}} \otimes ∣ j, r ⟩_{*}, = ∣ v ⟩_{B_{*}} \otimes i \sum ψ_{i} ∣ f, r, i ⟩_{*} .

Displayed formulas in this passage

Hawk01-B@1#eq:endpointenergy · line 1129
$0 \leq m_{*} < M - e_{K} - e .$
Hawk01-B@1#eq:endpointsuccess · line 1149
$V_{*} ∣ j, r ⟩_{B_{N}} V_{*} ∣ f, r; ψ ⟩_{B_{N}} = ∣ v ⟩_{B_{*}} \otimes ∣ j, r ⟩_{*}, = ∣ v ⟩_{B_{*}} \otimes i \sum ψ_{i} ∣ f, r, i ⟩_{*} .$
Hawk01-B@1#eq:endpointfailure · line 1149
$V_{*} ∣ j, r ⟩_{B_{N}} V_{*} ∣ f, r; ψ ⟩_{B_{N}} = ∣ v ⟩_{B_{*}} \otimes ∣ j, r ⟩_{*}, = ∣ v ⟩_{B_{*}} \otimes i \sum ψ_{i} ∣ f, r, i ⟩_{*} .$
unlabeled · line 1137
$E_{*}^{s} = M - e_{K} - e - m_{*} .$
unlabeled · line 1142
$E_{*}^{f} = M - e_{K} - m_{*} .$

Endpoint completion and the no-remnant condition

Displayed formulas in this passage

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