Manuscript XI-E@1 · XI-E.tex · section · 46e55514

The polynomial domain and product

Route 1: a named polynomial Hadamard extension › The polynomial domain and product · lines 714–736

XI-E@1#eq:star

This fixed-kernel polynomial class is sufficient here. Wick polynomials also belong to the more restrictive equicausal class discussed in HRV (Theorem 5.9).

Let $Γ_{H}$ be the graded contraction operator between two functional factors, normalized so that

Φ_{A} (u) ⋆_{H} Φ_{B} (v) = Φ_{A} (u) Φ_{B} (v) + ℏ ⟨ u, H_{A B} v ⟩ .

Define

F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G), E_{H} = (F_{mc, pol}^{sol} [[ℏ, κ]], ⋆_{H}) .

Each contraction series terminates for fixed polynomials. The graded derivative convention includes the Koszul signs for odd slots.

Displayed formulas in this passage

XI-E@1#eq:star · line 727
$F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G), E_{H} = (F_{mc, pol}^{sol} [[ℏ, κ]], ⋆_{H}) .$
unlabeled · line 722
$Φ_{A} (u) ⋆_{H} Φ_{B} (v) = Φ_{A} (u) Φ_{B} (v) + ℏ ⟨ u, H_{A B} v ⟩ .$

The polynomial domain and product

Displayed formulas in this passage

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