Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Exact prescribed-history bath overlap

Gaussian influence and a recovery obstruction · lines 1492–1538 · proof included or linked

Hawk02-E@1#prop:history

For real integrable histories $j_{a} (s)$ on a finite interval, consider the interaction-picture bath Hamiltonian

H_{j} (s) = a \sum j_{a} (s) Φ (f_{a, s}) .

Its propagator has the form

U_{j} (t) = e^{i θ_{j} (t)} W (η_{j} (t)), η_{j} (t) = - i a \sum \int_{0}^{t} j_{a} (s) f_{a, s} d s,

where

W (η) = exp (a^{†} (η) - a (η))

is the Weyl unitary and $θ_{j} (t)$ is real. For two histories,

⟨ Ω_{B}, U_{k} (t)^{†} U_{j} (t) Ω_{B} ⟩ = exp [- \frac{1}{2} ∥ η_{j} (t) - η_{k} (t) ∥^{2}] .

Proof. The commutator of $H_{j} (s)$ and $H_{j} (u)$ is a scalar multiple of the identity:

[Φ (f), Φ (g)] = 2 i Im ⟨ f, g ⟩ I .

Consequently chronological Weyl multiplication yields one Weyl displacement and a real common phase. The displacement follows by integrating the linear generator. The vacuum characteristic function is

⟨ Ω_{B}, W (η) Ω_{B} ⟩ = e^{- ∥ η ∥^{2} /2} .

The Weyl product phase has unit modulus, proving the overlap formula.

Displayed formulas in this passage

unlabeled · line 1496
$H_{j} (s) = a \sum j_{a} (s) Φ (f_{a, s}) .$
unlabeled · line 1500
$U_{j} (t) = e^{i θ_{j} (t)} W (η_{j} (t)), η_{j} (t) = - i a \sum \int_{0}^{t} j_{a} (s) f_{a, s} d s,$
unlabeled · line 1506
$W (η) = exp (a^{†} (η) - a (η))$
unlabeled · line 1512
$⟨ Ω_{B}, U_{k} (t)^{†} U_{j} (t) Ω_{B} ⟩ = exp [- \frac{1}{2} ∥ η_{j} (t) - η_{k} (t) ∥^{2}] .$
unlabeled · line 1524
$[Φ (f), Φ (g)] = 2 i Im ⟨ f, g ⟩ I .$
unlabeled · line 1532
$⟨ Ω_{B}, W (η) Ω_{B} ⟩ = e^{- ∥ η ∥^{2} /2} .$

Proposition: Exact prescribed-history bath overlap

Displayed formulas in this passage

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