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feat: Grading on FieldOpAlgebra

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jstoobysmith 2025-02-05 07:22:14 +00:00
parent 35445a5be6
commit 7d9e6af80c
10 changed files with 609 additions and 160 deletions
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18
HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
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@ -27,7 +27,7 @@ open FieldStatistic
map `𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra` such that
`superCommuteF (φ₀ᶜ…φₙᵃ) (φ₀'ᶜ…φₙ'ᶜ)` is equal to
`φ₀ᶜ…φₙᵃ * φ₀'ᶜ…φₙ'ᶜ - 𝓢(φ₀ᶜ…φₙᵃ, φ₀'ᶜ…φₙ'ᶜ) φ₀'ᶜ…φₙ'ᶜ * φ₀ᶜ…φₙᵃ`.
The notation `[a, b]ₛca` is used for this super commutator. -/
The notation `[a, b]ₛca` is used for this super commutator. -/
noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
𝓕.FieldOpFreeAlgebra :=
Basis.constr ofCrAnListFBasis ℂ fun φs =>
@ -637,7 +637,7 @@ lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj a]
rw [← bosonicProjF_add_fermionicProjF a]
simp only [map_add, LinearMap.add_apply]
rw [superCommuteF_bosonic_bosonic (by simp) hb, superCommuteF_fermionic_bonsonic (by simp) hb]
simp only [add_mul, mul_add]
@ -645,7 +645,7 @@ lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statist
lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj b]
rw [← bosonicProjF_add_fermionicProjF b]
simp only [map_add, LinearMap.add_apply]
rw [superCommuteF_bosonic_bosonic ha (by simp), superCommuteF_bosonic_fermionic ha (by simp)]
simp only [add_mul, mul_add]
@ -703,12 +703,12 @@ lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.FieldOpFreeAlgebra}
rw [superCommuteF_fermionic_fermionic hb ha]
abel
lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.FieldOpFreeAlgebra) :
[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]
lemma superCommuteF_expand_bosonicProjF_fermionicProjF (a b : 𝓕.FieldOpFreeAlgebra) :
[a, b]ₛca = bosonicProjF a * bosonicProjF b - bosonicProjF b * bosonicProjF a +
bosonicProjF a * fermionicProjF b - fermionicProjF b * bosonicProjF a +
fermionicProjF a * bosonicProjF b - bosonicProjF b * fermionicProjF a +
fermionicProjF a * fermionicProjF b + fermionicProjF b * fermionicProjF a := by
conv_lhs => rw [← bosonicProjF_add_fermionicProjF a, ← bosonicProjF_add_fermionicProjF b]
simp only [map_add, LinearMap.add_apply]
rw [superCommuteF_bonsonic (by simp),
superCommuteF_fermionic_bonsonic (by simp) (by simp),