Manuscript Em03-F@1 · Em03-F-v1.tex · section · 0d4d148b

The actual Hamiltonian path

The source and its exact compact interaction › The actual Hamiltonian path · lines 558–591

Em03-F@1#eq:HI

The lapse and spatial volume on a time slice are

N = f (r), d Σ_{t} = r^{2} f (r) d x d Ω.

Consequently

Φ_{F} (t) = \int_{Σ_{t}} N F (t, \cdot) ϕ (t, \cdot) d Σ_{t} = \int_{R} f (t, x) Ψ (t, x) d x,

and

\int Φ_{F} (t) d t = ϕ (F) .

The interaction-picture Hamiltonian is

H_{I} (t) = g Z_{S} \otimes Φ_{F} (t), g \in R .

The corresponding action term is

S_{int} = - g \int F (x) Z_{S} ϕ (x) d vol_{g} .

Define the instantaneous coefficient

k_{t, j} (ω) = \frac{e ^{i ω t}}{4 π ω} \int f (t, x) q_{j} (x, ω) d x .

Then $k_{F} = \int k_{t} d t = k (f)$ .

Displayed formulas in this passage

Em03-F@1#eq:HI · line 575
$H_{I} (t) = g Z_{S} \otimes Φ_{F} (t), g \in R .$
unlabeled · line 561
$N = f (r), d Σ_{t} = r^{2} f (r) d x d Ω.$
unlabeled · line 566
$Φ_{F} (t) = \int_{Σ_{t}} N F (t, \cdot) ϕ (t, \cdot) d Σ_{t} = \int_{R} f (t, x) Ψ (t, x) d x,$
unlabeled · line 571
$\int Φ_{F} (t) d t = ϕ (F) .$
unlabeled · line 580
$S_{int} = - g \int F (x) Z_{S} ϕ (x) d vol_{g} .$
unlabeled · line 585
$k_{t, j} (ω) = \frac{e ^{i ω t}}{4 π ω} \int f (t, x) q_{j} (x, ω) d x .$