Manuscript XIX-D@1 · XIX-D.tex · section · 1f47e33b

Formal remainders and regulator limits

Covariance, state selection, and quantum products › Formal remainders and regulator limits · lines 506–533

For a formal series

O = r, s \geq 0 \sum κ^{r} ℏ^{s} O_{r, s},

a statement modulo $κ^{R + 1}$ or $ℏ^{S + 1}$ is coefficientwise. It is not an estimate at numerical values of the parameters.

Likewise, an upper bound

∣ ⟨ O_{ϵ}, f ⟩ ∣ \leq C_{f} ϵ^{- r}

is neither convergence nor an evaluated divergence. The zero family and a family proportional to $ϵ^{- r}$ can satisfy the same upper bound.

The elementary family

Z_{ϵ} = ϵ^{- 1}

is smooth and invariant under every action on its configuration arguments, yet $⟨ Z_{ϵ}, f ⟩$ diverges for a test with nonzero integral. Thus classical covariance does not imply regulator removal.

Displayed formulas in this passage

unlabeled · line 509
$O = r, s \geq 0 \sum κ^{r} ℏ^{s} O_{r, s},$
unlabeled · line 517
$∣ ⟨ O_{ϵ}, f ⟩ ∣ \leq C_{f} ϵ^{- r}$
unlabeled · line 525
$Z_{ϵ} = ϵ^{- 1}$