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

Spaces, labels, and preparation

The autonomous source–boson Hamiltonian › Spaces, labels, and preparation · lines 534–590

Hawk02-C@2#eq:regularity, Hawk02-C@2#eq:H, Hawk02-C@2#eq:V

Let

H_{M} = span {∣ 0 ⟩, ∣ 1 ⟩, \dots, ∣ d ⟩}, J_{C} ∣ a ⟩_{C} = ∣ a ⟩_{M}, 1 \leq a \leq d,

and

P_{ex} = a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{M}, H_{M} = ω_{0} P_{ex}, ω_{0} > 0.

Define

k = L^{2} ((0, \infty), d ν), h = k \otimes C^{d} .

The field space is $F_{s} (h)$ , with vacuum $Ω$ . Multiplication by $ν$ is denoted $M_{ν}$ on either the scalar or labeled one-particle space, according to context.

The labels are distinct orthogonal field channels with identical dispersion and coupling. This engineered symmetry is an explicit model assumption, not a consequence of a local gravitational scalar field.

Let $L_{a} = ∣ 0 ⟩ ⟨ a ∣_{M}$ and $a_{a}^{†} (g) = a^{†} (g \otimes e_{a})$ . The inner product is conjugate linear in its first argument; annihilation is conjugate linear in its test. Assume

g \in k, κ_{g} := \int_{0}^{\infty} \frac{∣ g ( ν ) ∣ ^{2}}{ν} d ν < \infty.

The Hamiltonian is formally

H = H_{0} + V, H_{0} = ω_{0} P_{ex} \otimes I + I \otimes d Γ (M_{ν}),

V = a = 1 \sum^{d} (L_{a} \otimes a_{a}^{†} (g) + L_{a}^{†} \otimes a_{a} (g)) .

Its precise operator definition is sectorwise.

The prepared code isometry is

J ∣ a ⟩_{C} = ∣ a ⟩_{M} \otimes Ω.

No switching function or driving pulse enters the ideal evolution.

Displayed formulas in this passage

Hawk02-C@2#eq:regularity · line 566
$g \in k, κ_{g} := \int_{0}^{\infty} \frac{∣ g ( ν ) ∣ ^{2}}{ν} d ν < \infty.$
Hawk02-C@2#eq:H · line 572
$H = H_{0} + V, H_{0} = ω_{0} P_{ex} \otimes I + I \otimes d Γ (M_{ν}),$
Hawk02-C@2#eq:V · line 577
$V = a = 1 \sum^{d} (L_{a} \otimes a_{a}^{†} (g) + L_{a}^{†} \otimes a_{a} (g)) .$
unlabeled · line 537
$H_{M} = span {∣ 0 ⟩, ∣ 1 ⟩, \dots, ∣ d ⟩}, J_{C} ∣ a ⟩_{C} = ∣ a ⟩_{M}, 1 \leq a \leq d,$
unlabeled · line 543
$P_{ex} = a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{M}, H_{M} = ω_{0} P_{ex}, ω_{0} > 0.$
unlabeled · line 548
$k = L^{2} ((0, \infty), d ν), h = k \otimes C^{d} .$
unlabeled · line 586
$J ∣ a ⟩_{C} = ∣ a ⟩_{M} \otimes Ω.$