Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

An Part super Commute normal Order of Field Op List sum

FieldSpecification.WickAlgebra.anPart_superCommute_normalOrder_ofFieldOpList_sum

physics quantum field theory relativity lemmas

Plain-language statement

The commutator of the annihilation part of a field operator with a normal ordered list of field operators can be decomposed into the sum of the commutators of the annihilation part with each element of the list of field operators, i.e. [anPart φ, 𝓝(φ₀…φₙ)]ₛ= ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [anPart φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ).

Exact Lean statement

lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
    [anPart φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
    [anPart φ, ofFieldOpF φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :    [anPart φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •    [anPart φ, ofFieldOpF φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by  match φ with  | .inAsymp φ =>    simp only [anPart_inAsymp, map_zero, LinearMap.zero_apply, Fin.getElem_fin,      Algebra.smul_mul_assoc, zero_mul]    conv_rhs =>      enter [2, s]      rw [smul_zero]    simp  | .position φ =>    simp only [anPart_position, Fin.getElem_fin, Algebra.smul_mul_assoc]    rw [ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum]    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod,      Fin.getElem_fin, Algebra.smul_mul_assoc]    rfl  | .outAsymp φ =>    simp only [anPart_outAsymp, Fin.getElem_fin, Algebra.smul_mul_assoc]    rw [ofCrAnOp_superCommute_normalOrder_ofFieldOpList_sum]    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod,      Fin.getElem_fin, Algebra.smul_mul_assoc]    rfl

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean:297-319

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