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

Proposition: Individually defined distributions need not have a raw product

Covariance, state selection, and quantum products › The classical Hessian is not its contraction · lines 466–487 · proof included or linked

XIX-D@1#prop:product

There are smooth regularizations of two individually defined distributions whose ordinary products have no distributional limit.

Proof. Take a nonzero real $ρ \in C_{c}^{\infty} (R)$ with $\int ρ = 1$ , and put

δ_{ϵ} (t) = ϵ^{- 1} ρ (t / ϵ) .

Each $δ_{ϵ}$ converges to $δ_{0}$ . For a compact test $f$ with $f (0) > 0$ ,

⟨ δ_{ϵ}^{2}, f ⟩ = ϵ^{- 1} \int ρ (v)^{2} f (ϵ v) d v = ϵ^{- 1} f (0) \int ρ (v)^{2} d v + O (1) .

The leading coefficient is positive, so the pairing has no finite limit.

Displayed formulas in this passage

unlabeled · line 475
$δ_{ϵ} (t) = ϵ^{- 1} ρ (t / ϵ) .$
unlabeled · line 480
$⟨ δ_{ϵ}^{2}, f ⟩ = ϵ^{- 1} \int ρ (v)^{2} f (ϵ v) d v = ϵ^{- 1} f (0) \int ρ (v)^{2} d v + O (1) .$