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refactor: improve docs

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jstoobysmith 2025-01-24 11:09:25 +00:00
parent eeacdef74e
commit 2490535569
6 changed files with 126 additions and 57 deletions
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12
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
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@ -404,10 +404,14 @@ lemma superCommute_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List
rw [ofStateList_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
simp [mul_comm]
/--
Within the creation and annihilation algebra, we have that
`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
-/
lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
(φs' : List 𝓕.CrAnStates) →
[ofCrAnList φs, ofCrAnList φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ (List.take n φs')) •
(φs' : List 𝓕.CrAnStates) → [ofCrAnList φs, ofCrAnList φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofCrAnList (φs'.take n) * [ofCrAnList φs, ofCrAnState (φs'.get n)]ₛca *
ofCrAnList (φs'.drop (n + 1))
| [] => by
@ -422,7 +426,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
[ofCrAnList φs, ofStateList φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ List.take n φs') •
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofStateList (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
ofStateList (φs'.drop (n + 1))
| [] => by