Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

Accessible and residual spaces

Prescribed packet access and the complete residual › Accessible and residual spaces · lines 834–889

Hawk02-C@2#eq:AB, Hawk02-C@2#eq:r, Hawk02-C@2#eq:residualembedding

Fix a rank- $N$ orthogonal projection $P_{N}$ on $k$ , $N \geq 1$ , before the input is supplied. Use the same projection for every label:

Π_{N} = P_{N} \otimes I_{d} .

The canonical factorization is

F_{s} (h) ≅ F_{s} (Π_{N} h) \otimes F_{s} ((I - Π_{N}) h) .

Vacuum factorizes, and a one-particle vector separates into its two projected components.

The ideal assignment is

A_{N} = F_{s} (Π_{N} h), B_{N} = H_{M} \otimes F_{s} ((I - Π_{N}) h) .

This is declared subsystem access, not yet a local detector or compact spacetime apparatus.

At time $T$ , define

ℓ_{N} (T) = ∥ (I - P_{N}) f_{T} ∥_{2}^{2}, p_{N} (T) = ∥ P_{N} f_{T} ∥_{2}^{2},

r_{N} (T) = q (T) + ℓ_{N} (T) = 1 - p_{N} (T) .

Suppress $T$ temporarily.

Let

L_{N}^{col} ∣ a ⟩ = ∣ P_{N} f_{T}, a ⟩_{A_{N}},

and define the joint unnormalized residual embedding

R_{N} ∣ a ⟩ = A (T) ∣ a ⟩_{M} \otimes Ω_{miss} + ∣ 0 ⟩_{M} \otimes ∣(I - P_{N}) f_{T}, a ⟩_{miss} .

Set

a_{0} = Ω_{col}, b_{0} = ∣ 0 ⟩_{M} \otimes Ω_{miss} .

Displayed formulas in this passage

Hawk02-C@2#eq:AB · line 853
$A_{N} = F_{s} (Π_{N} h), B_{N} = H_{M} \otimes F_{s} ((I - Π_{N}) h) .$
Hawk02-C@2#eq:r · line 866
$r_{N} (T) = q (T) + ℓ_{N} (T) = 1 - p_{N} (T) .$
Hawk02-C@2#eq:residualembedding · line 877
$R_{N} ∣ a ⟩ = A (T) ∣ a ⟩_{M} \otimes Ω_{miss} + ∣ 0 ⟩_{M} \otimes ∣(I - P_{N}) f_{T}, a ⟩_{miss} .$
unlabeled · line 839
$Π_{N} = P_{N} \otimes I_{d} .$
unlabeled · line 843
$F_{s} (h) ≅ F_{s} (Π_{N} h) \otimes F_{s} ((I - Π_{N}) h) .$
unlabeled · line 862
$ℓ_{N} (T) = ∥ (I - P_{N}) f_{T} ∥_{2}^{2}, p_{N} (T) = ∥ P_{N} f_{T} ∥_{2}^{2},$
unlabeled · line 873
$L_{N}^{col} ∣ a ⟩ = ∣ P_{N} f_{T}, a ⟩_{A_{N}},$
unlabeled · line 885
$a_{0} = Ω_{col}, b_{0} = ∣ 0 ⟩_{M} \otimes Ω_{miss} .$