Manuscript XII-A@2 · XII-A-v2.tex · section · 955d4b22

The product and one contraction convention

The product and one contraction convention · lines 619–653

XII-A@2#eq:star, XII-A@2#eq:orientation

Use graded left derivatives and the graded tensor-product rule. Normalize the even contraction operator $Γ_{H}$ by

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) .

For homogeneous kernels of degrees $n, m$ , its $k$ -contraction is

\frac{ℏ ^{k}}{k !} \frac{n ! m !}{( n - k )! ( m - k )!} Sym_{n + m - 2 k} π_{*} [(u \otimes v) H^{\otimes k}], 0 \leq k \leq min (n, m) .

Here $π$ drops the paired variables; indices and Koszul signs are contracted as prescribed above.

The orientation convention is fixed once:

(y, p; y^{'}, - p) \in WF (H), p \in V_{σ} ⟹ q = - p \in V_{- σ}, q^{'} = p = - q

for covectors of $u, v$ canceled by that line. Zero entries occur when individual line factors are smooth. This convention is also used for the continuity stages.

Displayed formulas in this passage

XII-A@2#eq:star · line 629
$F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .$
XII-A@2#eq:orientation · line 644
$(y, p; y^{'}, - p) \in WF (H), p \in V_{σ} ⟹ q = - p \in V_{- σ}, q^{'} = p = - q$
unlabeled · line 624
$Φ_{A} (u) ⋆_{H} Φ_{B} (v) = Φ_{A} (u) Φ_{B} (v) + ℏ ⟨ u, H_{A B} v ⟩ .$
unlabeled · line 634
$\frac{ℏ ^{k}}{k !} \frac{n ! m !}{( n - k )! ( m - k )!} Sym_{n + m - 2 k} π_{*} [(u \otimes v) H^{\otimes k}], 0 \leq k \leq min (n, m) .$

The product and one contraction convention

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