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

Proposition: Two distinct finite-family structures

Finite families: spectral laws versus ordered Weyl correlations › Second moments and commutators · lines 867–916 · proof included or linked

Em06-B@1#prop:covariance, Em06-B@1#eq:cov-general, Em06-B@1#eq:comm-general

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For every finite-number residual state,

Im M_{ij}^{ρ} = A_{ij} .

The symmetric covariance in the displaced state is

Cov_{sym} (O_{i, ϵ}, O_{j, ϵ}) = ϵ C_{ij}^{ρ_{ϵ}} .

On the common core,

[O_{i, ϵ}, O_{j, ϵ}] = 2 i ϵ A_{ij} 1 .

For the displaced vacuum,

C_{ij}^{∣Ω ⟩ ⟨ Ω∣} = S_{ij},

so the symmetric covariance is exactly $ϵ S$ .

Proof. On finite-particle vectors the weak commutator gives

⟨ Φ (v_{i}) ψ, Φ (v_{j}) ψ ⟩ - ⟨ Φ (v_{j}) ψ, Φ (v_{i}) ψ ⟩ = 2 i A_{ij} ∥ ψ ∥^{2} .

Particle truncation converges in $Dom N^{1/2}$ , and (eq:number-bound) passes this identity to that form domain. Summing over the state decomposition proves the imaginary-part statement.

Displacement shifts each field by its real constant $o_{j}$ after scaling. Covariance is unaffected by this shift, proving (eq:cov-general). The core commutator is (eq:CCR) with the scale included. For the vacuum,

Φ (v_{j}) Ω = a^{†} (v_{j}) Ω, ⟨ Φ (v_{i}) Ω, Φ (v_{j}) Ω ⟩ = ⟨ v_{i}, v_{j} ⟩,

and all first moments vanish.

Displayed formulas in this passage

Em06-B@1#eq:cov-general · line 874
$Cov_{sym} (O_{i, ϵ}, O_{j, ϵ}) = ϵ C_{ij}^{ρ_{ϵ}} .$
Em06-B@1#eq:comm-general · line 881
$[O_{i, ϵ}, O_{j, ϵ}] = 2 i ϵ A_{ij} 1 .$
unlabeled · line 870
$Im M_{ij}^{ρ} = A_{ij} .$
unlabeled · line 887
$C_{ij}^{∣Ω ⟩ ⟨ Ω∣} = S_{ij},$
unlabeled · line 895
$⟨ Φ (v_{i}) ψ, Φ (v_{j}) ψ ⟩ - ⟨ Φ (v_{j}) ψ, Φ (v_{i}) ψ ⟩ = 2 i A_{ij} ∥ ψ ∥^{2} .$
unlabeled · line 910
$Φ (v_{j}) Ω = a^{†} (v_{j}) Ω, ⟨ Φ (v_{i}) Ω, Φ (v_{j}) Ω ⟩ = ⟨ v_{i}, v_{j} ⟩,$

Proposition: Two distinct finite-family structures

Displayed formulas in this passage

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