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

Lemma: Occupation-truncated dynamics

Reference-preserving finite approximation › Dynamics truncated at total occupation · lines 954–974 · proof included or linked

Hawk02-E@1#lem:occupation-dynamics, Hawk02-E@1#eq:occupation-dynamics

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For $0 \leq t \leq T$ ,

V_{t} - V_{t}^{(m)} \leq 2 Z_{m} (2 GT) .

Proof. The free evolution preserves particle number. In interaction picture, the exact and truncated Dyson terms acting on vacuum coincide through order $m$ : no intermediate state at those orders can exceed $m$ particles.

At every higher order, both terms satisfy the bound of Lemma lem:dyson; inserted projections are contractions. The difference is therefore bounded by twice the tail series. Returning to Schrödinger picture does not change the norm.

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Hawk02-E@1#eq:occupation-dynamics · line 957
$V_{t} - V_{t}^{(m)} \leq 2 Z_{m} (2 GT) .$

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section: Dynamics truncated at total occupation