feat: Redefine FieldOpAlgebra
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6 changed files with 469 additions and 23 deletions
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@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Grading
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/-!
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# Super Commute
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@ -438,6 +439,349 @@ lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) :
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· simp
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· simp [Finset.mul_sum, smul_smul, ofStateList_cons, mul_assoc,
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FieldStatistic.ofList_cons_eq_mul, mul_comm]
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/-!
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## Interaction with grading.
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-/
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lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
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(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
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[a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
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[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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simp [p]
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
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[a2 , ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs', rfl, hφs'⟩ := hx
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList]
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apply Submodule.sub_mem _
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· apply ofCrAnList_mem_statisticSubmodule_of
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rw [ofList_append_eq_mul, hφs, hφs']
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· apply Submodule.smul_mem
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apply ofCrAnList_mem_statisticSubmodule_of
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rw [ofList_append_eq_mul, hφs, hφs']
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rw [mul_comm]
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp [p]
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exact Submodule.add_mem _ hp1 hp2
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· intro c x hx hp1
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simp [p]
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exact Submodule.smul_mem _ c hp1
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· exact ha
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp [p]
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exact Submodule.add_mem _ hp1 hp2
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· intro c x hx hp1
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simp [p]
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exact Submodule.smul_mem _ c hp1
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· exact hb
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lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs', rfl, hφs'⟩ := hx
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp [hφs, ofCrAnList_append]
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact ha
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs', rfl, hφs'⟩ := hx
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp [hφs, hφs', ofCrAnList_append]
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact ha
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
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[a, a2]ₛca = a * a2 - a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs', rfl, hφs'⟩ := hx
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp [hφs, hφs', ofCrAnList_append]
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact ha
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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rw [← bosonicProj_add_fermionicProj a]
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simp only [map_add, LinearMap.add_apply]
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rw [superCommute_bosonic_bosonic (by simp) hb, superCommute_fermionic_bonsonic (by simp) hb]
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simp only [add_mul, mul_add]
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abel
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lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = a * b - b * a := by
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rw [← bosonicProj_add_fermionicProj b]
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simp only [map_add, LinearMap.add_apply]
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rw [superCommute_bosonic_bosonic ha (by simp), superCommute_bosonic_fermionic ha (by simp)]
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simp only [add_mul, mul_add]
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abel
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lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = - [b, a]ₛca := by
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rw [bosonic_superCommute hb, superCommute_bonsonic hb]
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simp
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lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
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[a, b]ₛca = - [b, a]ₛca := by
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rw [bosonic_superCommute ha, superCommute_bonsonic ha]
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simp
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lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
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[a, b]ₛca = a * b + b * a := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a, a2]ₛca = a * a2 + a2 * a
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change p b hb
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs, rfl, hφs⟩ := hx
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
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[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro x hx
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obtain ⟨φs', rfl, hφs'⟩ := hx
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp [hφs, hφs', ofCrAnList_append]
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact ha
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· simp [p]
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· intro x y hx hy hp1 hp2
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simp_all [p, mul_add, add_mul]
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abel
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· intro c x hx hp1
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simp_all [p, smul_sub]
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· exact hb
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lemma superCommute_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
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(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
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[a, b]ₛca = [b, a]ₛca := by
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rw [superCommute_fermionic_fermionic ha hb]
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rw [superCommute_fermionic_fermionic hb ha]
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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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by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
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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
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apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
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apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
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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
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apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
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apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
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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
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apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
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apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
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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
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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_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
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(ha : a ∈ statisticSubmodule bosonic) :
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1)) := by
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let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1))
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro a ha
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obtain ⟨φs, rfl, hφs⟩ := ha
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList_eq_sum]
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congr
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funext n
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simp [hφs]
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· simp [p]
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· intro x y hx hy hpx hpy
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simp_all [p]
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rw [← Finset.sum_add_distrib]
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congr
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funext n
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simp [mul_add, add_mul]
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· intro c x hx hpx
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simp_all [p, Finset.smul_sum]
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· exact ha
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lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
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(ha : a ∈ statisticSubmodule fermionic) :
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1)) := by
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let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
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[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
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ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
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ofCrAnList (φs.drop (n + 1))
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change p a ha
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apply Submodule.span_induction (p := p)
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· intro a ha
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obtain ⟨φs, rfl, hφs⟩ := ha
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simp [p]
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rw [superCommute_ofCrAnList_ofCrAnList_eq_sum]
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congr
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funext n
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simp [hφs]
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· simp [p]
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· intro x y hx hy hpx hpy
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simp_all [p]
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rw [← Finset.sum_add_distrib]
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congr
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funext n
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simp [mul_add, add_mul]
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· intro c x hx hpx
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simp_all [p, Finset.smul_sum]
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congr
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funext x
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simp [smul_smul, mul_comm]
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· exact ha
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lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
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(h : [ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic) :
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(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnList φs, ofCrAnList φs']ₛca = 0 := by
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by_cases h0 : [ofCrAnList φs, ofCrAnList φs']ₛca = 0
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· simp [h0]
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simp [h0]
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by_contra hn
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refine h0 (eq_zero_of_bosonic_and_fermionic ?_ h)
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by_cases hc : (𝓕 |>ₛ φs) = bosonic
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· have h1 : bosonic = bosonic + bosonic := by
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simp
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rfl
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rw [h1]
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apply superCommute_grade
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apply ofCrAnList_mem_statisticSubmodule_of _ _ hc
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apply ofCrAnList_mem_statisticSubmodule_of _ _
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rw [← hn, hc]
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· have h1 : bosonic = fermionic + fermionic := by
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simp
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rfl
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rw [h1]
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apply superCommute_grade
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apply ofCrAnList_mem_statisticSubmodule_of _ _
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simpa using hc
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apply ofCrAnList_mem_statisticSubmodule_of _ _
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rw [← hn]
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simpa using hc
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end CrAnAlgebra
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