refactor: Update supercommute notation
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jstoobysmith
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12 changed files with 197 additions and 188 deletions
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@ -24,24 +24,24 @@ variable (𝓕 : FieldSpecification)
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def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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{ x |
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(∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
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x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca)
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x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF)
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∨ (∃ (φc φc' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
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x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛca)
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x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛF)
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∨ (∃ (φa φa' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
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x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛca)
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x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛF)
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∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
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x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
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x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛF)}
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/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
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of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by
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- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` field creation operators.
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- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛF` for `φc` and `φc'` field creation operators.
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This corresponds to the condition that two creation operators always super-commute.
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- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` field annihilation operators.
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- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛF` for `φa` and `φa'` field annihilation operators.
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This corresponds to the condition that two annihilation operators always super-commute.
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- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
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- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛF` for `φ` and `φ'` operators with different statistics.
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This corresponds to the condition that two operators with different statistics always
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super-commute. In other words, fermions and bosons always super-commute.
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- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
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- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF`. This corresponds to the condition,
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when combined with the conditions above, that the super-commutator is in the center
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of the algebra.
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-/
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@ -70,7 +70,7 @@ lemma equiv_iff_exists_add (x y : FieldOpFreeAlgebra 𝓕) :
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rw [equiv_iff_sub_mem_ideal]
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simp [ha]
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/-- For a field specification `𝓕`, the projection
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/-- For a field specification `𝓕`, `ι` is defined as the projection
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`𝓕.FieldOpFreeAlgebra →ₐ[ℂ] FieldOpAlgebra 𝓕`
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@ -100,7 +100,7 @@ lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.f
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simpa using hx
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lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = .create)
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(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛca = 0 := by
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(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛF = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
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simp only [exists_prop]
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@ -110,7 +110,7 @@ lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnFieldOp) (hφc :
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lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
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(hφa : 𝓕 |>ᶜ φa = .annihilate) (hφa' : 𝓕 |>ᶜ φa' = .annihilate) :
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ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛca = 0 := by
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ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛF = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
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simp only [exists_prop]
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@ -120,7 +120,7 @@ lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
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use φa, φa', hφa, hφa'
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lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnFieldOp}
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
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right
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@ -129,8 +129,8 @@ lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnFieldOp}
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use φ, ψ
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lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnFieldOp)
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(h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule fermionic) :
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
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(h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ statisticSubmodule fermionic) :
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton] at h ⊢
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rcases statistic_neq_of_superCommuteF_fermionic h with h | h
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· simp only [ofCrAnListF_singleton]
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@ -139,8 +139,8 @@ lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnFieldOp)
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· simp [h]
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lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnFieldOp) :
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[ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule bosonic ∨
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
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[ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ statisticSubmodule bosonic ∨
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
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rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [ψ] with h | h
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· simp_all [ofCrAnListF_singleton]
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· simp_all only [ofCrAnListF_singleton]
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@ -155,14 +155,14 @@ lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnFie
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@[simp]
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lemma ι_superCommuteF_ofCrAnOpF_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
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ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca = 0 := by
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ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
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left
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use φ1, φ2, φ3
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lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
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ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnOpF φ3]ₛca = 0 := by
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ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, ofCrAnOpF φ3]ₛF = 0 := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
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rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
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· rw [bonsonic_superCommuteF_symm h]
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@ -175,7 +175,7 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3
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lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 : 𝓕.CrAnFieldOp)
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(φs : List 𝓕.CrAnFieldOp) :
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ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnListF φs]ₛca = 0 := by
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ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, ofCrAnListF φs]ₛF = 0 := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
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rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
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· rw [superCommuteF_bosonic_ofCrAnListF_eq_sum _ _ h]
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@ -185,27 +185,27 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 :
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@[simp]
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lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnFieldOp)
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(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca = 0 := by
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change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) a = _
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have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) = 0 := by
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(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, a]ₛF = 0 := by
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change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF) a = _
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have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF) = 0 := by
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apply (ofCrAnListFBasis.ext fun l ↦ ?_)
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simp [ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF]
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rw [h1]
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simp
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lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 : 𝓕.CrAnFieldOp)
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(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca -
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ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca * ι a = 0 := by
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(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF -
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ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF * ι a = 0 := by
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rcases ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero φ1 φ2 with h | h
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swap
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· simp [h]
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trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca
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trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, a]ₛF
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· rw [bosonic_superCommuteF h]
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simp
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· simp
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lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnFieldOp) :
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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rw [Subalgebra.mem_center_iff]
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intro a
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obtain ⟨a, rfl⟩ := ι_surjective a
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