feat: Add SuperCommute for FieldOpAlgebra
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jstoobysmith
parent
c2d89cc093
commit
9107115620
5 changed files with 708 additions and 9 deletions
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@ -37,6 +37,11 @@ lemma ofCrAnList_bosonic_or_fermionic (φs : List 𝓕.CrAnStates) :
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· right
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exact ofCrAnList_mem_statisticSubmodule_of φs fermionic (by simpa using h)
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lemma ofCrAnState_bosonic_or_fermionic (φ : 𝓕.CrAnStates) :
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ofCrAnState φ ∈ statisticSubmodule bosonic ∨ ofCrAnState φ ∈ statisticSubmodule fermionic := by
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rw [← ofCrAnList_singleton]
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exact ofCrAnList_bosonic_or_fermionic [φ]
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/-- The projection of an element of `CrAnAlgebra` onto it's bosonic part. -/
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def bosonicProj : 𝓕.CrAnAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) bosonic :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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@ -183,6 +188,7 @@ lemma bosonicProj_add_fermionicProj (a : 𝓕.CrAnAlgebra) :
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· simp [h]
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· simp [h]
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lemma coeAddMonoidHom_apply_eq_bosonic_plus_fermionic
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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DirectSum.coeAddMonoidHom statisticSubmodule a = a.1 bosonic + a.1 fermionic := by
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@ -229,7 +235,7 @@ lemma directSum_eq_bosonic_plus_fermionic
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conv_lhs => rw [hx, hy]
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abel
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instance : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
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instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
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one_mem := by
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simp only [statisticSubmodule]
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refine Submodule.mem_span.mpr fun p a => a ?_
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@ -296,6 +302,100 @@ lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.CrAnAlgebra}
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rw [ha, hb] at hc
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simpa using hc
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lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
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(a * b).bosonicProj.1 = a.bosonicProj.1 * b.bosonicProj.1
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+ a.fermionicProj.1 * b.fermionicProj.1 := by
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conv_lhs =>
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rw [← bosonicProj_add_fermionicProj a]
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rw [← bosonicProj_add_fermionicProj b]
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simp [mul_add, add_mul]
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rw [bosonicProj_of_mem_bosonic]
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conv_lhs =>
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left
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right
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rw [bosonicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = fermionic + bosonic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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left
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rw [bosonicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = bosonic + fermionic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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right
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rw [bosonicProj_of_mem_bosonic _
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(by
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have h1 : bosonic = fermionic + fermionic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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simp
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· have h1 : bosonic = bosonic + bosonic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp
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lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
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(a * b).fermionicProj.1 = a.bosonicProj.1 * b.fermionicProj.1
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+ a.fermionicProj.1 * b.bosonicProj.1 := by
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conv_lhs =>
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rw [← bosonicProj_add_fermionicProj a]
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rw [← bosonicProj_add_fermionicProj b]
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simp [mul_add, add_mul]
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conv_lhs =>
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left
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left
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rw [fermionicProj_of_mem_bosonic _
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(by
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have h1 : bosonic = bosonic + bosonic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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left
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right
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rw [fermionicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = fermionic + bosonic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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left
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rw [fermionicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = bosonic + fermionic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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right
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rw [fermionicProj_of_mem_bosonic _
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(by
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have h1 : bosonic = fermionic + fermionic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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simp
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abel
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end
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@ -688,6 +688,18 @@ lemma superCommute_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
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rw [superCommute_fermionic_fermionic hb ha]
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abel
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lemma superCommute_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
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[a, b]ₛca = bosonicProj a * bosonicProj b - bosonicProj b * bosonicProj a +
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bosonicProj a * fermionicProj b - fermionicProj b * bosonicProj a +
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fermionicProj a * bosonicProj b - bosonicProj b * fermionicProj a +
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fermionicProj a * fermionicProj b + fermionicProj b * fermionicProj a := by
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conv_lhs => rw [← bosonicProj_add_fermionicProj a, ← bosonicProj_add_fermionicProj b]
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simp
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rw [superCommute_bonsonic (by simp), superCommute_bosonic_fermionic (by simp) (by simp),
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superCommute_fermionic_bonsonic (by simp) (by simp),
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superCommute_fermionic_fermionic (by simp) (by simp)]
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abel
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lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
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@ -725,6 +737,41 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
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apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
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apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
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lemma superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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exact superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
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lemma superCommute_superCommute_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
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[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
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[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
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rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ2 φ3 with hs | hs
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<;> rcases ofCrAnState_bosonic_or_fermionic φ1 with h1 | h1
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· left
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have h : bosonic = bosonic + bosonic := by
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simp only [add_eq_mul, instCommGroup, mul_self]
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rfl
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rw [h]
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apply superCommute_grade h1 hs
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· right
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have h : fermionic = fermionic + bosonic := by
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simp only [add_eq_mul, instCommGroup, mul_self]
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rfl
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rw [h]
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apply superCommute_grade h1 hs
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· right
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have h : fermionic = bosonic + fermionic := by
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simp only [add_eq_mul, instCommGroup, mul_self]
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rfl
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rw [h]
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apply superCommute_grade h1 hs
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· left
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have h : bosonic = fermionic + fermionic := by
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simp only [add_eq_mul, instCommGroup, mul_self]
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rfl
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rw [h]
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apply superCommute_grade h1 hs
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lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
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(ha : a ∈ statisticSubmodule bosonic) :
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@ -191,5 +191,241 @@ lemma ι_superCommute_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnState
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rw [← h0]
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abel
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/-!
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## The kernal of ι
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-/
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lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
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ι x = 0 ↔ x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [ι_apply]
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change ⟦x⟧ = ⟦0⟧ ↔ _
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simp only [ringConGen, Quotient.eq]
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rw [TwoSidedIdeal.mem_iff]
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simp only
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rfl
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lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
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have hx' := hx
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simp [fieldOpIdealSet] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
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have hx' := hx
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simp [fieldOpIdealSet] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
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let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
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a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
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change p x hx
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apply AddSubgroup.closure_induction
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· intro x hx
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simp [p]
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obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
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obtain ⟨d, hd, y, hy, rfl⟩ := Set.mem_mul.mp ha
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rw [bosonicProj_mul, bosonicProj_mul, fermionicProj_mul]
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simp [mul_add, add_mul]
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rcases fermionicProj_mem_fieldOpIdealSet_or_zero y hy with hfy | hfy
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<;> rcases bosonicProj_mem_fieldOpIdealSet_or_zero y hy with hby | hby
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· apply TwoSidedIdeal.add_mem
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apply TwoSidedIdeal.add_mem
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· /- boson, boson, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (bosonicProj y).1
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simp [hby]
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· use ↑(bosonicProj b)
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simp
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(bosonicProj b)
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simp
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apply TwoSidedIdeal.add_mem
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· /- boson, fermion, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· /- fermion, boson, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (bosonicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· simp [hby]
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apply TwoSidedIdeal.add_mem
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(bosonicProj b)
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simp
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· /- boson, fermion, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
|
||||
· simp [hfy]
|
||||
apply TwoSidedIdeal.add_mem
|
||||
· /- boson, boson, boson mem-/
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||||
apply Set.mem_mul.mpr
|
||||
use ↑(bosonicProj d) * ↑(bosonicProj y)
|
||||
apply And.intro
|
||||
· apply Set.mem_mul.mpr
|
||||
use bosonicProj d
|
||||
simp
|
||||
use (bosonicProj y).1
|
||||
simp [hby]
|
||||
· use ↑(bosonicProj b)
|
||||
simp
|
||||
· /- fermion, boson, fermion mem-/
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||||
apply Set.mem_mul.mpr
|
||||
use ↑(fermionicProj d) * ↑(bosonicProj y)
|
||||
apply And.intro
|
||||
· apply Set.mem_mul.mpr
|
||||
use fermionicProj d
|
||||
simp
|
||||
use (bosonicProj y).1
|
||||
simp [hby, hfy]
|
||||
· use ↑(fermionicProj b)
|
||||
simp
|
||||
· simp [hfy, hby]
|
||||
· simp [p]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [p]
|
||||
apply TwoSidedIdeal.add_mem
|
||||
exact hpx
|
||||
exact hpy
|
||||
· intro x hx
|
||||
simp [p]
|
||||
|
||||
lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||||
x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||||
have hb := bosonicProj_mem_ideal x hx
|
||||
rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢
|
||||
rw [← bosonicProj_add_fermionicProj x] at hx
|
||||
simp at hx
|
||||
simp_all
|
||||
|
||||
lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
|
||||
ι x = 0 ↔ ι x.bosonicProj.1 = 0 ∧ ι x.fermionicProj.1 = 0 := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
rw [@ι_eq_zero_iff_mem_ideal] at h ⊢
|
||||
rw [ι_eq_zero_iff_mem_ideal]
|
||||
apply And.intro
|
||||
· exact bosonicProj_mem_ideal x h
|
||||
· exact fermionicProj_mem_ideal x h
|
||||
· intro h
|
||||
rw [← bosonicProj_add_fermionicProj x]
|
||||
simp_all
|
||||
|
||||
|
||||
|
||||
end FieldOpAlgebra
|
||||
end FieldSpecification
|
||||
|
|
|
|||
|
|
@ -0,0 +1,114 @@
|
|||
/-
|
||||
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.TimeOrder
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
|
||||
/-!
|
||||
|
||||
# SuperCommute on Field operator algebra
|
||||
|
||||
-/
|
||||
|
||||
namespace FieldSpecification
|
||||
open CrAnAlgebra
|
||||
open HepLean.List
|
||||
open FieldStatistic
|
||||
|
||||
namespace FieldOpAlgebra
|
||||
variable {𝓕 : FieldSpecification}
|
||||
|
||||
lemma ι_superCommute_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
|
||||
ι [a, b]ₛca = 0 := by
|
||||
rw [superCommute_expand_bosonicProj_fermionicProj]
|
||||
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
|
||||
simp_all
|
||||
|
||||
lemma ι_superCommute_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
|
||||
ι [a, b]ₛca = 0 := by
|
||||
rw [superCommute_expand_bosonicProj_fermionicProj]
|
||||
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
|
||||
simp_all
|
||||
|
||||
/-!
|
||||
|
||||
## Defining normal order for `FiedOpAlgebra`.
|
||||
|
||||
-/
|
||||
|
||||
lemma ι_superCommute_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
|
||||
(h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
|
||||
apply ι_superCommute_eq_zero_of_ι_right_zero
|
||||
exact (ι_eq_zero_iff_mem_ideal b).mpr h
|
||||
|
||||
lemma ι_superCommute_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
|
||||
ι [a, b1]ₛca = ι [a, b2]ₛca := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||||
simp only [LinearMap.mem_ker, ← map_sub]
|
||||
exact ι_superCommute_right_zero_of_mem_ideal a (b1 - b2) h
|
||||
|
||||
noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a) (ι_superCommute_eq_of_equiv_right a)
|
||||
map_add' x y := by
|
||||
obtain ⟨x, hx⟩ := ι_surjective x
|
||||
obtain ⟨y, hy⟩ := ι_surjective y
|
||||
subst hx hy
|
||||
rw [← map_add, ι_apply, ι_apply, ι_apply]
|
||||
rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
|
||||
simp
|
||||
map_smul' c y := by
|
||||
obtain ⟨y, hy⟩ := ι_surjective y
|
||||
subst hy
|
||||
rw [← map_smul, ι_apply, ι_apply]
|
||||
simp
|
||||
|
||||
lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) : superCommuteRight a (ι b) = ι [a, b]ₛca := by
|
||||
rfl
|
||||
|
||||
lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) : superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by
|
||||
rfl
|
||||
|
||||
lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
|
||||
superCommuteRight a1 = superCommuteRight a2 := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
ext b
|
||||
obtain ⟨b, rfl⟩ := ι_surjective b
|
||||
have ha1b1 : (superCommuteRight (a1 - a2)) (ι b) = 0 := by
|
||||
rw [superCommuteRight_apply_ι]
|
||||
apply ι_superCommute_eq_zero_of_ι_left_zero
|
||||
exact (ι_eq_zero_iff_mem_ideal (a1 - a2)).mpr h
|
||||
simp_all [superCommuteRight_apply_ι]
|
||||
trans ι ((superCommute a2) b) + 0
|
||||
rw [← ha1b1]
|
||||
simp
|
||||
simp
|
||||
|
||||
noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
|
||||
FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift superCommuteRight superCommuteRight_eq_of_equiv
|
||||
map_add' x y := by
|
||||
obtain ⟨x, rfl⟩ := ι_surjective x
|
||||
obtain ⟨y, rfl⟩ := ι_surjective y
|
||||
ext b
|
||||
obtain ⟨b, rfl⟩ := ι_surjective b
|
||||
rw [← map_add, ι_apply, ι_apply, ι_apply, ι_apply]
|
||||
rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
|
||||
simp
|
||||
rw [superCommuteRight_apply_quot, superCommuteRight_apply_quot, superCommuteRight_apply_quot]
|
||||
simp
|
||||
map_smul' c y := by
|
||||
obtain ⟨y, rfl⟩ := ι_surjective y
|
||||
ext b
|
||||
obtain ⟨b, rfl⟩ := ι_surjective b
|
||||
rw [← map_smul, ι_apply, ι_apply, ι_apply]
|
||||
simp
|
||||
rw [superCommuteRight_apply_quot, superCommuteRight_apply_quot]
|
||||
simp
|
||||
|
||||
lemma ι_superCommute (a b : 𝓕.CrAnAlgebra) : ι [a, b]ₛca = superCommute (ι a) (ι b) := by
|
||||
rfl
|
||||
|
||||
end FieldOpAlgebra
|
||||
end FieldSpecification
|
||||
|
|
@ -19,7 +19,164 @@ open FieldStatistic
|
|||
namespace FieldOpAlgebra
|
||||
variable {𝓕 : FieldSpecification}
|
||||
|
||||
lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
|
||||
lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
|
||||
(φs1 φs2 : List 𝓕.CrAnStates) (h :
|
||||
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
|
||||
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
|
||||
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2):
|
||||
ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
|
||||
= 0 := by
|
||||
let l1 :=
|
||||
(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
|
||||
++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs1)
|
||||
let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
|
||||
++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
|
||||
have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
|
||||
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
|
||||
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)):= by
|
||||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
|
||||
(by simp_all)
|
||||
rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2 by simp,
|
||||
crAnTimeOrderList, h1]
|
||||
simp only [List.append_assoc, List.singleton_append, decide_not,
|
||||
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
|
||||
have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
|
||||
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
|
||||
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)):= by
|
||||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
|
||||
(by simp_all)
|
||||
rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2 by simp,
|
||||
crAnTimeOrderList, h1]
|
||||
simp only [List.singleton_append, decide_not,
|
||||
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
|
||||
congr 1
|
||||
have hp : List.Perm [φ1, φ3, φ2] [φ1, φ2, φ3] := by
|
||||
refine List.Perm.cons φ1 ?_
|
||||
exact List.Perm.swap φ2 φ3 []
|
||||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
simp_all
|
||||
have hp231 : List.Perm [φ2, φ3, φ1] [φ1, φ2, φ3] := by
|
||||
refine List.Perm.trans (l₂ := [φ2, φ1, φ3]) ?_ ?_
|
||||
refine List.Perm.cons φ2 (List.Perm.swap φ1 φ3 [])
|
||||
exact List.Perm.swap φ1 φ2 [φ3]
|
||||
have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
|
||||
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
|
||||
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)):= by
|
||||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
|
||||
(by simp_all)
|
||||
rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2 by simp,
|
||||
crAnTimeOrderList, h1]
|
||||
simp only [List.singleton_append, decide_not,
|
||||
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
|
||||
congr 1
|
||||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp231, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
simp_all
|
||||
have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
|
||||
crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
|
||||
• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)):= by
|
||||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
|
||||
(by simp_all)
|
||||
rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2 by simp,
|
||||
crAnTimeOrderList, h1]
|
||||
simp only [List.singleton_append, decide_not,
|
||||
Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
|
||||
congr 1
|
||||
have hp : List.Perm [φ3, φ2, φ1] [φ1, φ2, φ3] := by
|
||||
refine List.Perm.trans ?_ hp231
|
||||
exact List.Perm.swap φ2 φ3 [φ1]
|
||||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
simp_all
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||||
simp
|
||||
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
|
||||
simp [mul_sub, sub_mul, ← ofCrAnList_append]
|
||||
rw [h123, h132, h231, h321]
|
||||
simp [smul_smul]
|
||||
rw [mul_comm, ← smul_smul, mul_comm, ← smul_smul]
|
||||
rw [← smul_sub, ← smul_sub, smul_smul, mul_comm, ← smul_smul, ← smul_sub]
|
||||
simp
|
||||
right
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm, mul_assoc]
|
||||
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm]
|
||||
rw [smul_sub]
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm]
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm]
|
||||
repeat rw [mul_assoc]
|
||||
rw [← mul_sub, ← mul_sub, ← mul_sub]
|
||||
rw [← sub_mul, ← sub_mul, ← sub_mul]
|
||||
trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ι (ofCrAnList l2)
|
||||
rw [mul_assoc]
|
||||
congr
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||||
simp
|
||||
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
|
||||
simp [smul_sub]
|
||||
simp_all
|
||||
|
||||
lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
|
||||
(φs1 φs2 : List 𝓕.CrAnStates):
|
||||
ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
|
||||
= 0 := by
|
||||
by_cases h :
|
||||
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
|
||||
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
|
||||
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2
|
||||
· exact ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList φs1 φs2 h
|
||||
· rw [timeOrder_timeOrder_mid]
|
||||
rw [timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel _ _ _ h]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra):
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
|
||||
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
|
||||
change pb b (Basis.mem_span _ b)
|
||||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs, rfl⟩ := hx
|
||||
simp [pb]
|
||||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
|
||||
change pa a (Basis.mem_span _ a)
|
||||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl⟩ := hx
|
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simp [pa]
|
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exact ι_timeOrder_superCommute_superCommute_ofCrAnList φs' φs
|
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· simp [pa]
|
||||
· intro x y hx hy hpx hpy
|
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simp_all [pa,mul_add, add_mul]
|
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· intro x hx hpx
|
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simp_all [pa, hpx]
|
||||
· simp [pb]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [pb,mul_add, add_mul]
|
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· intro x hx hpx
|
||||
simp_all [pb, hpx]
|
||||
|
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lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
|
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
|
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
|
||||
|
|
@ -110,6 +267,25 @@ lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
|
|||
· intro x hx hpx
|
||||
simp_all [pb, hpx]
|
||||
|
||||
|
||||
lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
|
||||
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
|
||||
rw [timeOrder_timeOrder_mid]
|
||||
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
|
||||
exact Decidable.not_and_iff_or_not.mp hφψ
|
||||
rcases hφψ with hφψ | hφψ
|
||||
· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
|
||||
have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
|
||||
simp_all
|
||||
simp_all
|
||||
· rw [superCommute_ofCrAnState_ofCrAnState_symm]
|
||||
simp
|
||||
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
|
||||
simp
|
||||
simp_all
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Defining normal order for `FiedOpAlgebra`.
|
||||
|
|
@ -130,16 +306,42 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
match hc with
|
||||
| Or.inl hc =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
simp
|
||||
| Or.inr (Or.inl hc) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
rw [ι_superCommute_of_create_create]
|
||||
simp
|
||||
· exact hφa
|
||||
· exact hφb
|
||||
· exact heqt.1
|
||||
· exact heqt.2
|
||||
· rw [ι_timeOrder_superCommute_neq_time heqt]
|
||||
| Or.inr (Or.inr (Or.inl hc)) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
rw [ι_superCommute_of_annihilate_annihilate]
|
||||
simp
|
||||
· exact hφa
|
||||
· exact hφb
|
||||
· exact heqt.1
|
||||
· exact heqt.2
|
||||
· rw [ι_timeOrder_superCommute_neq_time heqt]
|
||||
| Or.inr (Or.inr (Or.inr hc)) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
obtain ⟨φa, φb, hdiff, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
rw [ι_superCommute_of_diff_statistic]
|
||||
simp
|
||||
· exact hdiff
|
||||
· exact heqt.1
|
||||
· exact heqt.2
|
||||
· rw [ι_timeOrder_superCommute_neq_time heqt]
|
||||
· simp [p]
|
||||
· intro x y hx hy
|
||||
simp only [map_add, p]
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue