443 lines
21 KiB
Text
443 lines
21 KiB
Text
/-
|
||
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.Basic
|
||
/-!
|
||
|
||
# Super Commute
|
||
-/
|
||
|
||
namespace FieldSpecification
|
||
variable {𝓕 : FieldSpecification}
|
||
|
||
namespace CrAnAlgebra
|
||
|
||
open StateAlgebra
|
||
|
||
/-!
|
||
|
||
## The super commutor on the CrAnAlgebra.
|
||
|
||
-/
|
||
|
||
open FieldStatistic
|
||
|
||
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
|
||
or a bosonic and fermionic operator this corresponds to the usual commutator
|
||
whilst for two fermionic operators this corresponds to the anti-commutator. -/
|
||
noncomputable def superCommute : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
|
||
Basis.constr ofCrAnListBasis ℂ fun φs =>
|
||
Basis.constr ofCrAnListBasis ℂ fun φs' =>
|
||
ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs)
|
||
|
||
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
|
||
or a bosonic and fermionic operator this corresponds to the usual commutator
|
||
whilst for two fermionic operators this corresponds to the anti-commutator. -/
|
||
scoped[FieldSpecification.CrAnAlgebra] notation "⟨" φs "," φs' "⟩ₛca" => superCommute φs φs'
|
||
|
||
/-!
|
||
|
||
## The super commutor of different types of elements
|
||
|
||
-/
|
||
|
||
lemma superCommute_ofCrAnList_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
|
||
⟨ofCrAnList φs, ofCrAnList φs'⟩ₛca =
|
||
ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs) := by
|
||
rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
|
||
simp only [superCommute, Basis.constr_basis]
|
||
|
||
lemma superCommute_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
|
||
⟨ofCrAnState φ, ofCrAnState φ'⟩ₛca =
|
||
ofCrAnState φ * ofCrAnState φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnState φ' * ofCrAnState φ := by
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||
rw [superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append]
|
||
congr
|
||
rw [ofCrAnList_append]
|
||
rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
|
||
|
||
lemma superCommute_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
|
||
⟨ofCrAnList φcas, ofStateList φs⟩ₛca = ofCrAnList φcas * ofStateList φs -
|
||
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofStateList φs * ofCrAnList φcas := by
|
||
conv_lhs => rw [ofStateList_sum]
|
||
rw [map_sum]
|
||
conv_lhs =>
|
||
enter [2, x]
|
||
rw [superCommute_ofCrAnList_ofCrAnList, CrAnSection.statistics_eq_state_statistics,
|
||
ofCrAnList_append, ofCrAnList_append]
|
||
rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.smul_sum,
|
||
← Finset.sum_mul, ← ofStateList_sum]
|
||
simp
|
||
|
||
lemma superCommute_ofStateList_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
|
||
⟨ofStateList φ, ofStateList φs⟩ₛca = ofStateList φ * ofStateList φs -
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofStateList φ := by
|
||
conv_lhs => rw [ofStateList_sum]
|
||
simp only [map_sum, LinearMap.coeFn_sum, Finset.sum_apply, instCommGroup.eq_1,
|
||
Algebra.smul_mul_assoc]
|
||
conv_lhs =>
|
||
enter [2, x]
|
||
rw [superCommute_ofCrAnList_ofStatesList]
|
||
simp only [instCommGroup.eq_1, CrAnSection.statistics_eq_state_statistics,
|
||
Algebra.smul_mul_assoc, Finset.sum_sub_distrib]
|
||
rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofStateList_sum]
|
||
|
||
lemma superCommute_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||
⟨ofState φ, ofStateList φs⟩ₛca = ofState φ * ofStateList φs -
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofState φ := by
|
||
rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
|
||
simp
|
||
|
||
lemma superCommute_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
|
||
⟨ofStateList φs, ofState φ⟩ₛca = ofStateList φs * ofState φ -
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs := by
|
||
rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
|
||
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
|
||
match φ, φ' with
|
||
| States.inAsymp φ, _ =>
|
||
simp
|
||
| _, States.outAsymp φ =>
|
||
simp only [crPart_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
|
||
sub_self]
|
||
| States.position φ, States.position φ' =>
|
||
simp only [anPart_position, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.outAsymp φ, States.position φ' =>
|
||
simp only [anPart_posAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.position φ, States.inAsymp φ' =>
|
||
simp only [anPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp only [List.singleton_append, instCommGroup.eq_1, crAnStatistics,
|
||
FieldStatistic.ofList_singleton, Function.comp_apply, crAnStatesToStates_prod, ←
|
||
ofCrAnList_append]
|
||
| States.outAsymp φ, States.inAsymp φ' =>
|
||
simp only [anPart_posAsymp, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
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') -
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||
anPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) := by
|
||
match φ, φ' with
|
||
| States.outAsymp φ, _ =>
|
||
simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
|
||
mul_zero, sub_self]
|
||
| _, States.inAsymp φ =>
|
||
simp only [anPart_negAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
|
||
sub_self]
|
||
| States.position φ, States.position φ' =>
|
||
simp only [crPart_position, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.position φ, States.outAsymp φ' =>
|
||
simp only [crPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.inAsymp φ, States.position φ' =>
|
||
simp only [crPart_negAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.inAsymp φ, States.outAsymp φ' =>
|
||
simp only [crPart_negAsymp, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
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') -
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||
crPart (StateAlgebra.ofState φ') * crPart (StateAlgebra.ofState φ) := by
|
||
match φ, φ' with
|
||
| States.outAsymp φ, _ =>
|
||
simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
|
||
mul_zero, sub_self]
|
||
| _, States.outAsymp φ =>
|
||
simp only [crPart_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul, sub_self]
|
||
| States.position φ, States.position φ' =>
|
||
simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.position φ, States.inAsymp φ' =>
|
||
simp only [crPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.inAsymp φ, States.position φ' =>
|
||
simp only [crPart_negAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.inAsymp φ, States.inAsymp φ' =>
|
||
simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
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') -
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||
anPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ) := by
|
||
match φ, φ' with
|
||
| States.inAsymp φ, _ =>
|
||
simp
|
||
| _, States.inAsymp φ =>
|
||
simp
|
||
| States.position φ, States.position φ' =>
|
||
simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.position φ, States.outAsymp φ' =>
|
||
simp only [anPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.outAsymp φ, States.position φ' =>
|
||
simp only [anPart_posAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [crAnStatistics, ← ofCrAnList_append]
|
||
| States.outAsymp φ, States.outAsymp φ' =>
|
||
simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
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
|
||
match φ with
|
||
| States.inAsymp φ =>
|
||
simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
|
||
simp [crAnStatistics]
|
||
| States.position φ =>
|
||
simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
|
||
simp [crAnStatistics]
|
||
| States.outAsymp φ =>
|
||
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
|
||
match φ with
|
||
| States.inAsymp φ =>
|
||
simp
|
||
| States.position φ =>
|
||
simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
|
||
simp [crAnStatistics]
|
||
| States.outAsymp φ =>
|
||
simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
|
||
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
|
||
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
|
||
rw [← ofStateList_singleton, superCommute_anPart_ofStateList]
|
||
simp
|
||
|
||
/-!
|
||
|
||
## Mul equal superCommute
|
||
|
||
Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
|
||
multiplication with a sign plus the super commutor.
|
||
|
||
-/
|
||
lemma ofCrAnList_mul_ofCrAnList_eq_superCommute (φs φs' : List 𝓕.CrAnStates) :
|
||
ofCrAnList φs * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnList φs
|
||
+ ⟨ofCrAnList φs, ofCrAnList φs'⟩ₛca := by
|
||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||
simp [ofCrAnList_append]
|
||
|
||
lemma ofCrAnState_mul_ofCrAnList_eq_superCommute (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
|
||
ofCrAnState φ * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnState φ
|
||
+ ⟨ofCrAnState φ, ofCrAnList φs'⟩ₛca := by
|
||
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommute]
|
||
simp
|
||
|
||
lemma ofStateList_mul_ofStateList_eq_superCommute (φs φs' : List 𝓕.States) :
|
||
ofStateList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofStateList φs
|
||
+ ⟨ofStateList φs, ofStateList φs'⟩ₛca := by
|
||
rw [superCommute_ofStateList_ofStatesList]
|
||
simp
|
||
|
||
lemma ofState_mul_ofStateList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
|
||
ofState φ * ofStateList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofStateList φs' * ofState φ
|
||
+ ⟨ofState φ, ofStateList φs'⟩ₛca := by
|
||
rw [superCommute_ofState_ofStatesList]
|
||
simp
|
||
|
||
lemma ofStateList_mul_ofState_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
|
||
ofStateList φs * ofState φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs
|
||
+ ⟨ofStateList φs, ofState φ⟩ₛca := by
|
||
rw [superCommute_ofStateList_ofState]
|
||
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
|
||
rw [superCommute_crPart_anPart]
|
||
simp
|
||
|
||
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
|
||
anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState 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
|
||
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
|
||
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
|
||
rw [superCommute_anPart_anPart]
|
||
simp
|
||
|
||
lemma ofCrAnList_mul_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||
ofCrAnList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofCrAnList φs
|
||
+ ⟨ofCrAnList φs, ofStateList φs'⟩ₛca := by
|
||
rw [superCommute_ofCrAnList_ofStatesList]
|
||
simp
|
||
|
||
/-!
|
||
|
||
## Symmetry of the super commutor.
|
||
|
||
-/
|
||
|
||
lemma superCommute_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates) :
|
||
⟨ofCrAnList φs, ofCrAnList φs'⟩ₛca =
|
||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • ⟨ofCrAnList φs', ofCrAnList φs⟩ₛca := by
|
||
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList, smul_sub]
|
||
simp only [instCommGroup.eq_1, neg_smul, sub_neg_eq_add]
|
||
rw [smul_smul]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [exchangeSign_symm, exchangeSign_mul_self]
|
||
simp only [one_smul]
|
||
abel
|
||
|
||
lemma superCommute_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
|
||
⟨ofCrAnState φ, ofCrAnState φ'⟩ₛca =
|
||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • ⟨ofCrAnState φ', ofCrAnState φ⟩ₛca := by
|
||
rw [superCommute_ofCrAnState_ofCrAnState, superCommute_ofCrAnState_ofCrAnState]
|
||
rw [smul_sub]
|
||
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, neg_smul, sub_neg_eq_add]
|
||
rw [smul_smul]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [exchangeSign_symm, exchangeSign_mul_self]
|
||
simp only [one_smul]
|
||
abel
|
||
|
||
/-!
|
||
|
||
## Splitting the super commutor on lists into sums.
|
||
|
||
-/
|
||
lemma superCommute_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
|
||
⟨ofCrAnList φs, ofCrAnList (φ :: φs')⟩ₛca =
|
||
⟨ofCrAnList φs, ofCrAnState φ⟩ₛca * ofCrAnList φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
|
||
• ofCrAnState φ * ⟨ofCrAnList φs, ofCrAnList φs'⟩ₛca := by
|
||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||
conv_rhs =>
|
||
lhs
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList, sub_mul, ← ofCrAnList_append]
|
||
rhs
|
||
rw [FieldStatistic.ofList_singleton, ofCrAnList_append, ofCrAnList_singleton, smul_mul_assoc,
|
||
mul_assoc, ← ofCrAnList_append]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [superCommute_ofCrAnList_ofCrAnList, mul_sub, smul_mul_assoc]
|
||
simp only [instCommGroup.eq_1, List.cons_append, List.append_assoc, List.nil_append,
|
||
Algebra.mul_smul_comm, Algebra.smul_mul_assoc, sub_add_sub_cancel, sub_right_inj]
|
||
rw [← ofCrAnList_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||
simp only [instCommGroup, map_mul, mul_comm]
|
||
|
||
lemma superCommute_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
|
||
(φs' : List 𝓕.States) : ⟨ofCrAnList φs, ofStateList (φ :: φs')⟩ₛca =
|
||
⟨ofCrAnList φs, ofState φ⟩ₛca * ofStateList φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ⟨ofCrAnList φs, ofStateList φs'⟩ₛca := by
|
||
rw [superCommute_ofCrAnList_ofStatesList]
|
||
conv_rhs =>
|
||
lhs
|
||
rw [← ofStateList_singleton, superCommute_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
|
||
← ofStateList_append]
|
||
rhs
|
||
rw [FieldStatistic.ofList_singleton, ofStateList_singleton, smul_mul_assoc,
|
||
smul_mul_assoc, mul_assoc]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [superCommute_ofCrAnList_ofStatesList, mul_sub, smul_mul_assoc]
|
||
simp only [instCommGroup, Algebra.smul_mul_assoc, List.singleton_append, Algebra.mul_smul_comm,
|
||
sub_add_sub_cancel, sub_right_inj]
|
||
rw [ofStateList_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||
simp [mul_comm]
|
||
|
||
lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
|
||
(φs' : List 𝓕.CrAnStates) →
|
||
⟨ofCrAnList φs, ofCrAnList φs'⟩ₛca =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ (List.take n φs')) •
|
||
ofCrAnList (φs'.take n) * ⟨ofCrAnList φs, ofCrAnState (φs'.get n)⟩ₛca *
|
||
ofCrAnList (φs'.drop (n + 1))
|
||
| [] => by
|
||
simp [← ofCrAnList_nil, superCommute_ofCrAnList_ofCrAnList]
|
||
| φ :: φs' => by
|
||
rw [superCommute_ofCrAnList_ofCrAnList_cons, superCommute_ofCrAnList_ofCrAnList_eq_sum φs φs']
|
||
conv_rhs => erw [Fin.sum_univ_succ]
|
||
congr 1
|
||
· simp
|
||
· simp [Finset.mul_sum, smul_smul, ofCrAnList_cons, mul_assoc,
|
||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||
|
||
lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
|
||
⟨ofCrAnList φs, ofStateList φs'⟩ₛca =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ List.take n φs') •
|
||
ofStateList (φs'.take n) * ⟨ofCrAnList φs, ofState (φs'.get n)⟩ₛca *
|
||
ofStateList (φs'.drop (n + 1))
|
||
| [] => by
|
||
simp only [superCommute_ofCrAnList_ofStatesList, instCommGroup, ofList_empty,
|
||
exchangeSign_bosonic, one_smul, List.length_nil, Finset.univ_eq_empty, List.take_nil,
|
||
List.get_eq_getElem, List.drop_nil, Finset.sum_empty]
|
||
simp
|
||
| φ :: φs' => by
|
||
rw [superCommute_ofCrAnList_ofStateList_cons, superCommute_ofCrAnList_ofStateList_eq_sum φs φs']
|
||
conv_rhs => erw [Fin.sum_univ_succ]
|
||
congr 1
|
||
· simp
|
||
· simp [Finset.mul_sum, smul_smul, ofStateList_cons, mul_assoc,
|
||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||
|
||
end CrAnAlgebra
|
||
|
||
end FieldSpecification
|