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feat: Time order for CrAnAlgebra

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Also remove StateAlgebra
This commit is contained in:
jstoobysmith 2025-01-27 11:12:48 +00:00
parent 3abc31af98
commit 21f81a9331
19 changed files with 493 additions and 290 deletions
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72
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
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@ -14,7 +14,6 @@ variable {𝓕 : FieldSpecification}
namespace CrAnAlgebra
open StateAlgebra
/-!
@ -97,9 +96,9 @@ lemma superCommute_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.State
simp
lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
[anPart (StateAlgebra.ofState φ), crPart (StateAlgebra.ofState φ')]ₛca =
anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ) := by
[anPart φ, crPart φ']ₛca =
anPart φ * crPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
simp
@ -126,10 +125,10 @@ lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
[crPart (StateAlgebra.ofState φ), anPart (StateAlgebra.ofState φ')]ₛca =
crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') -
[crPart φ, anPart φ']ₛca =
crPart φ * anPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
anPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) := by
anPart φ' * crPart φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
@ -155,10 +154,10 @@ lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) :
[crPart (StateAlgebra.ofState φ), crPart (StateAlgebra.ofState φ')]ₛca =
crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') -
[crPart φ, crPart φ']ₛca =
crPart φ * crPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
crPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) := by
crPart φ' * crPart φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
@ -183,10 +182,10 @@ lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) :
[anPart (StateAlgebra.ofState φ), anPart (StateAlgebra.ofState φ')]ₛca =
anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') -
[anPart φ, anPart φ']ₛca =
anPart φ * anPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
anPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ) := by
anPart φ' * anPart φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
simp
@ -210,9 +209,9 @@ lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) :
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
[crPart (StateAlgebra.ofState φ), ofStateList φs]ₛca =
crPart (StateAlgebra.ofState φ) * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs *
crPart (StateAlgebra.ofState φ) := by
[crPart φ, ofStateList φs]ₛca =
crPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs *
crPart φ := by
match φ with
| States.inAsymp φ =>
simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
@ -226,9 +225,9 @@ lemma superCommute_crPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States
simp
lemma superCommute_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
[anPart (StateAlgebra.ofState φ), ofStateList φs]ₛca =
anPart (StateAlgebra.ofState φ) * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
ofStateList φs * anPart (StateAlgebra.ofState φ) := by
[anPart φ, ofStateList φs]ₛca =
anPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
ofStateList φs * anPart φ := by
match φ with
| States.inAsymp φ =>
simp
@ -242,16 +241,16 @@ lemma superCommute_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States
simp [crAnStatistics]
lemma superCommute_crPart_ofState (φ φ' : 𝓕.States) :
[crPart (StateAlgebra.ofState φ), ofState φ']ₛca =
crPart (StateAlgebra.ofState φ) * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPart (StateAlgebra.ofState φ) := by
[crPart φ, ofState φ']ₛca =
crPart φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPart φ := by
rw [← ofStateList_singleton, superCommute_crPart_ofStateList]
simp
lemma superCommute_anPart_ofState (φ φ' : 𝓕.States) :
[anPart (StateAlgebra.ofState φ), ofState φ']ₛca =
anPart (StateAlgebra.ofState φ) * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPart (StateAlgebra.ofState φ) := by
[anPart φ, ofState φ']ₛca =
anPart φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPart φ := by
rw [← ofStateList_singleton, superCommute_anPart_ofStateList]
simp
@ -294,31 +293,30 @@ lemma ofStateList_mul_ofState_eq_superCommute (φs : List 𝓕.States) (φ :
simp
lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) +
[crPart (StateAlgebra.ofState φ), anPart (StateAlgebra.ofState φ')]ₛca := by
crPart φ * anPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ +
[crPart φ, anPart φ']ₛca := by
rw [superCommute_crPart_anPart]
simp
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') =
anPart φ * crPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ) +
[anPart (StateAlgebra.ofState φ), crPart (StateAlgebra.ofState φ')]ₛca := by
crPart φ' * anPart φ +
[anPart φ, crPart φ']ₛca := by
rw [superCommute_anPart_crPart]
simp
lemma crPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) +
[crPart (StateAlgebra.ofState φ), crPart (StateAlgebra.ofState φ')]ₛca := by
crPart φ * crPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ +
[crPart φ, crPart φ']ₛca := by
rw [superCommute_crPart_crPart]
simp
lemma anPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ) +
[anPart (StateAlgebra.ofState φ), anPart (StateAlgebra.ofState φ')]ₛca := by
anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ +
[anPart φ, anPart φ']ₛca := by
rw [superCommute_anPart_anPart]
simp