Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Of Cr An Op super Commute normal Order of Cr An List sum

FieldSpecification.WickAlgebra.ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum

physics quantum field theory relativity lemmas

Plain-language statement

For a field specification 𝓕, an element φ of 𝓕.CrAnFieldOp, a list φs of 𝓕.CrAnFieldOp, the following relation holds [φ, 𝓝(φ₀…φₙ)]ₛ = ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ). The proof of this result ultimately goes as follows - The definition of normalOrder is used to rewrite 𝓝(φ₀…φₙ) as a scalar multiple of a `ofCrAnList...

Exact Lean statement

lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)
    (φs : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, 𝓝(ofCrAnList φs)]ₛ = ∑ n : Fin φs.length,
    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [ofCrAnOp φ, ofCrAnOp φs[n]]ₛ
    * 𝓝(ofCrAnList (φs.eraseIdx n))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)    (φs : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, 𝓝(ofCrAnList φs)]ₛ = ∑ n : Fin φs.length,    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [ofCrAnOp φ, ofCrAnOp φs[n]]ₛ    * 𝓝(ofCrAnList (φs.eraseIdx n)) := by  rw [normalOrder_ofCrAnList, map_smul]  erw [superCommute_ofCrAnOp_ofCrAnList_eq_sum, Finset.smul_sum,    sum_normalOrderList_length]  congr  funext n  simp only [List.get_eq_getElem, normalOrderList_get_normalOrderEquiv,    normalOrderList_eraseIdx_normalOrderEquiv, Algebra.smul_mul_assoc, Fin.getElem_fin]  erw [ofCrAnList_eq_normalOrder, mul_smul_comm, smul_smul, smul_smul]  by_cases hs : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[n])  · congr    erw [normalOrderSign_eraseIdx, ← hs]    trans (normalOrderSign φs * normalOrderSign φs) *      (𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv n))) *      𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv n))))      * 𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ (φs.take n))    · ring_nf      rw [hs]      rfl    · simp [hs]  · erw [superCommute_diff_statistic hs]    simp only [zero_mul, smul_zero]

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean:244-268

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Deriv Within mean Energy Beta eq neg variance

CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance

Plain-language statement

(∂U/∂β) = -Var(E) for finite systems.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record