Manuscript IV@1 · IV.tex · lemma · 351cb2d7

Lemma: Scalar symplectic correspondence

Model 1: the fixed vacuum TT plane › Three scalar one-particle vectors · lines 968–983 · proof included or linked

IV@1#lem:scalarCCR

For real $g, h$ ,

2 Im ⟨ u_{g}, u_{h} ⟩ = \int (α_{g} β_{h} - β_{g} α_{h}) d x .

Thus $ℓ_{g} \mapsto Φ (u_{g})$ has the canonical commutator for scalar initial linear functionals.

Proof. Expand (eq:ug). The two diagonal terms have real inner products for real data. Plancherel's identity identifies the imaginary cross terms with half the displayed expression.

Displayed formulas in this passage

unlabeled · line 971
$2 Im ⟨ u_{g}, u_{h} ⟩ = \int (α_{g} β_{h} - β_{g} α_{h}) d x .$

Nearby in this section

section: Three scalar one-particle vectors
section: Three scalar one-particle vectors