Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

The polynomial distribution carrier

Fixed spacetime, fields, and kernel carrier › The polynomial distribution carrier · lines 450–486

IX-A@1#eq:polynomial, IX-A@1#eq:coefficient-topology

Let

E = S^{2} T^{*} M \oplus (M \times C), E = C^{\infty} (M; E) .

A configuration is denoted by $Ψ = (h, ϕ)$ .

Let $P_{dist}$ be the space of finite polynomial functionals

F (Ψ) = n = 0 \sum^{N} \frac{1}{n !} ⟨ u_{n}, Ψ^{\otimes n} ⟩, u_{n} \in E^{'} (M^{n}; (E^{⊠ n})^{*}),

with symmetric kernels. Here $u_{0} \in C$ . Tensor indices are paired in the indicated bundles.

The compact distribution $u_{n}$ acts on smooth functions by inserting a compact cutoff equal to one near its support. This does not depend on the cutoff.

Equip the coefficient carrier with

T_{coeff} = n \geq 0 ⨁ E^{'} (M^{n}; (E^{⊠ n})^{*})_{strong},

using the locally convex direct sum and the strong dual topology of $C^{\infty}$ in each degree.

Displayed formulas in this passage

IX-A@1#eq:polynomial · line 460
$F (Ψ) = n = 0 \sum^{N} \frac{1}{n !} ⟨ u_{n}, Ψ^{\otimes n} ⟩, u_{n} \in E^{'} (M^{n}; (E^{⊠ n})^{*}),$
IX-A@1#eq:coefficient-topology · line 477
$T_{coeff} = n \geq 0 ⨁ E^{'} (M^{n}; (E^{⊠ n})^{*})_{strong},$
unlabeled · line 453
$E = S^{2} T^{*} M \oplus (M \times C), E = C^{\infty} (M; E) .$

Nearby in this section

lemma: Lemma: Uniqueness of polynomial kernels