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feat: Properties of super commute

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jstoobysmith 2025-01-30 07:16:19 +00:00
parent f7e669910c
commit c18b4850e5
3 changed files with 357 additions and 13 deletions
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21
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
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@ -96,8 +96,7 @@ lemma superCommuteF_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.Stat
simp
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
[anPartF φ, crPartF φ']ₛca =
anPartF φ * crPartF φ' -
[anPartF φ, crPartF φ']ₛca = anPartF φ * crPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
@ -125,10 +124,8 @@ lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
[crPartF φ, anPartF φ']ₛca =
crPartF φ * anPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
anPartF φ' * crPartF φ := by
[crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
@ -154,10 +151,8 @@ lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
[crPartF φ, crPartF φ']ₛca =
crPartF φ * crPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
crPartF φ' * crPartF φ := by
[crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
@ -183,9 +178,7 @@ lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.States) :
[anPartF φ, anPartF φ']ₛca =
anPartF φ * anPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
anPartF φ' * anPartF φ := by
anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
simp
@ -362,6 +355,7 @@ lemma superCommuteF_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
## Splitting the super commutor on lists into sums.
-/
lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList (φ :: φs')]ₛca =
[ofCrAnList φs, ofCrAnState φ]ₛca * ofCrAnList φs' +
@ -483,6 +477,7 @@ lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
simp only [h1, h2, h3, mul_self, map_one, one_smul, fermionic_exchangeSign_fermionic, neg_smul,
neg_sub]
abel
/-!
## Interaction with grading.