Manuscript Em06-B@1 · Em06-B-v1.tex · section · 61e38859

The free scalar carrier and unbounded moments

The free scalar carrier and unbounded moments · lines 645–676

Em06-B@1#eq:displacement, Em06-B@1#eq:Weyl, Em06-B@1#eq:CCR

We always mean these self-adjoint closures. Commutator identities for unbounded fields are stated on the invariant common core, or as the corresponding quadratic-form identities, not as everywhere-defined products.

Define the unitary displacement and scaled Weyl operators by

D (z) W_{ϵ} (v) = exp (a^{†} (z) - a (z)), = exp (i ϵ Φ (v)), 0 < ϵ \leq 1.

Their conventions give

D (z)^{*} Φ (v) D (z) W_{ϵ} (v) W_{ϵ} (w) [Φ (v), Φ (w)] = Φ (v) + 2 Re ⟨ v, z ⟩ 1, = e^{- i ϵ Im ⟨ v, w ⟩} W_{ϵ} (v + w), = 2 i Im ⟨ v, w ⟩ 1 on D_{fin} .

These follow by the scalar commutators of the generators, first on analytic vectors and then for the bounded unitary groups. They are the standard displacement/Weyl identities Glauber, with the signs fixed by our inner-product convention.

Displayed formulas in this passage

unlabeled · line 653
$D (z) W_{ϵ} (v) = exp (a^{†} (z) - a (z)), = exp (i ϵ Φ (v)), 0 < ϵ \leq 1.$
Em06-B@1#eq:displacement · line 659
$D (z)^{*} Φ (v) D (z) W_{ϵ} (v) W_{ϵ} (w) [Φ (v), Φ (w)] = Φ (v) + 2 Re ⟨ v, z ⟩ 1, = e^{- i ϵ Im ⟨ v, w ⟩} W_{ϵ} (v + w), = 2 i Im ⟨ v, w ⟩ 1 on D_{fin} .$
Em06-B@1#eq:Weyl · line 659
$D (z)^{*} Φ (v) D (z) W_{ϵ} (v) W_{ϵ} (w) [Φ (v), Φ (w)] = Φ (v) + 2 Re ⟨ v, z ⟩ 1, = e^{- i ϵ Im ⟨ v, w ⟩} W_{ϵ} (v + w), = 2 i Im ⟨ v, w ⟩ 1 on D_{fin} .$
Em06-B@1#eq:CCR · line 659
$D (z)^{*} Φ (v) D (z) W_{ϵ} (v) W_{ϵ} (w) [Φ (v), Φ (w)] = Φ (v) + 2 Re ⟨ v, z ⟩ 1, = e^{- i ϵ Im ⟨ v, w ⟩} W_{ϵ} (v + w), = 2 i Im ⟨ v, w ⟩ 1 on D_{fin} .$

The free scalar carrier and unbounded moments

Displayed formulas in this passage

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