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

A convergent vacuum Dyson expansion

Reference-preserving finite approximation › A convergent vacuum Dyson expansion · lines 864–887

Hawk02-E@1#eq:Z-tail

In the interaction picture with respect to $H_{S} + H_{B}$ , write

V_{I} (s) = a \sum g_{a} A_{a} (s) \otimes Φ (f_{a, s}),

where

A_{a} (s) = e^{i s H_{S}} A_{a} e^{- i s H_{S}}, f_{a, s} = e^{i s h_{B}} f_{a} .

These transformations preserve the norms in (eq:G).

For $x \geq 0$ , define

Z_{m} (x) = ℓ = m + 1 \sum^{\infty} \frac{x ^{ℓ}}{ℓ !} .

This series is finite for every finite $x$ .

Displayed formulas in this passage

Hawk02-E@1#eq:Z-tail · line 880
$Z_{m} (x) = ℓ = m + 1 \sum^{\infty} \frac{x ^{ℓ}}{ℓ !} .$
unlabeled · line 867
$V_{I} (s) = a \sum g_{a} A_{a} (s) \otimes Φ (f_{a, s}),$
unlabeled · line 872
$A_{a} (s) = e^{i s H_{S}} A_{a} e^{- i s H_{S}}, f_{a, s} = e^{i s h_{B}} f_{a} .$