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feat: Add SuperCommute for FieldOpAlgebra

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jstoobysmith 2025-01-29 15:08:43 +00:00
parent c2d89cc093
commit 9107115620
5 changed files with 708 additions and 9 deletions
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47
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
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@ -688,6 +688,18 @@ lemma superCommute_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
rw [superCommute_fermionic_fermionic hb ha]
abel
lemma superCommute_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
[a, b]ₛca = bosonicProj a * bosonicProj b - bosonicProj b * bosonicProj a +
bosonicProj a * fermionicProj b - fermionicProj b * bosonicProj a +
fermionicProj a * bosonicProj b - bosonicProj b * fermionicProj a +
fermionicProj a * fermionicProj b + fermionicProj b * fermionicProj a := by
conv_lhs => rw [← bosonicProj_add_fermionicProj a, ← bosonicProj_add_fermionicProj b]
simp
rw [superCommute_bonsonic (by simp), superCommute_bosonic_fermionic (by simp) (by simp),
superCommute_fermionic_bonsonic (by simp) (by simp),
superCommute_fermionic_fermionic (by simp) (by simp)]
abel
lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
@ -725,6 +737,41 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
lemma superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
exact superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
lemma superCommute_superCommute_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ2 φ3 with hs | hs
<;> rcases ofCrAnState_bosonic_or_fermionic φ1 with h1 | h1
· left
have h : bosonic = bosonic + bosonic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
· right
have h : fermionic = fermionic + bosonic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
· right
have h : fermionic = bosonic + fermionic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
· left
have h : bosonic = fermionic + fermionic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule bosonic) :