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refactor: Rename States to FieldOps

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jstoobysmith 2025-02-03 11:28:14 +00:00
parent 171e80fc04
commit 8f41de5785
36 changed files with 946 additions and 946 deletions
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90
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Basic.lean
View file

@ -3,7 +3,7 @@ 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.FieldSpecification.CrAnStates
import HepLean.PerturbationTheory.FieldSpecification.CrAnFieldOp
import HepLean.PerturbationTheory.FieldSpecification.CrAnSection
/-!
@ -35,63 +35,63 @@ namespace FieldSpecification
variable {𝓕 : FieldSpecification}
/-- The creation and annihlation free-algebra.
The free algebra generated by `CrAnStates`,
The free algebra generated by `CrAnFieldOp`,
that is a position based states or assymptotic states with a specification of
whether the state is a creation or annihlation state.
As a module `FieldOpFreeAlgebra` is spanned by lists of `CrAnStates`. -/
abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnStates
As a module `FieldOpFreeAlgebra` is spanned by lists of `CrAnFieldOp`. -/
abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnFieldOp
namespace FieldOpFreeAlgebra
/-- Maps a creation and annihlation state to the creation and annihlation free-algebra. -/
def ofCrAnOpF (φ : 𝓕.CrAnStates) : FieldOpFreeAlgebra 𝓕 :=
def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
FreeAlgebra.ι ℂ φ
/-- Maps a list creation and annihlation state to the creation and annihlation free-algebra
by taking their product. -/
def ofCrAnListF (φs : List 𝓕.CrAnStates) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
@[simp]
lemma ofCrAnListF_nil : ofCrAnListF ([] : List 𝓕.CrAnStates) = 1 := rfl
lemma ofCrAnListF_nil : ofCrAnListF ([] : List 𝓕.CrAnFieldOp) = 1 := rfl
lemma ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates) :
lemma ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs : List 𝓕.CrAnFieldOp) :
ofCrAnListF (φ :: φs) = ofCrAnOpF φ * ofCrAnListF φs := rfl
lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnStates) :
lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
ofCrAnListF (φs ++ φs') = ofCrAnListF φs * ofCrAnListF φs' := by
simp [ofCrAnListF, List.map_append]
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnStates) :
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
/-- Maps a state to the sum of creation and annihilation operators in
creation and annihilation free-algebra. -/
def ofFieldOpF (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.statesToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
/-- Maps a list of states to the creation and annihilation free-algebra by taking
the product of their sums of creation and annihlation operators.
Roughly `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)` etc. -/
def ofFieldOpListF (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
/-- Coercion from `List 𝓕.States` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
@[simp]
lemma ofFieldOpListF_nil : ofFieldOpListF ([] : List 𝓕.States) = 1 := rfl
lemma ofFieldOpListF_nil : ofFieldOpListF ([] : List 𝓕.FieldOp) = 1 := rfl
lemma ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
ofFieldOpListF (φ :: φs) = ofFieldOpF φ * ofFieldOpListF φs := rfl
lemma ofFieldOpListF_singleton (φ : 𝓕.States) :
lemma ofFieldOpListF_singleton (φ : 𝓕.FieldOp) :
ofFieldOpListF [φ] = ofFieldOpF φ := by simp [ofFieldOpListF]
lemma ofFieldOpListF_append (φs φs' : List 𝓕.States) :
lemma ofFieldOpListF_append (φs φs' : List 𝓕.FieldOp) :
ofFieldOpListF (φs ++ φs') = ofFieldOpListF φs * ofFieldOpListF φs' := by
dsimp only [ofFieldOpListF]
rw [List.map_append, List.prod_append]
lemma ofFieldOpListF_sum (φs : List 𝓕.States) :
lemma ofFieldOpListF_sum (φs : List 𝓕.FieldOp) :
ofFieldOpListF φs = ∑ (s : CrAnSection φs), ofCrAnListF s.1 := by
induction φs with
| nil => simp
@ -113,60 +113,60 @@ lemma ofFieldOpListF_sum (φs : List 𝓕.States) :
/-- The algebra map taking an element of the free-state algbra to
the part of it in the creation and annihlation free algebra
spanned by creation operators. -/
def crPartF : 𝓕.States → 𝓕.FieldOpFreeAlgebra := fun φ =>
def crPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
match φ with
| States.inAsymp φ => ofCrAnOpF ⟨States.inAsymp φ, ()⟩
| States.position φ => ofCrAnOpF ⟨States.position φ, CreateAnnihilate.create⟩
| States.outAsymp _ => 0
| FieldOp.inAsymp φ => ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩
| FieldOp.position φ => ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩
| FieldOp.outAsymp _ => 0
@[simp]
lemma crPartF_negAsymp (φ : 𝓕.IncomingAsymptotic) :
crPartF (States.inAsymp φ) = ofCrAnOpF ⟨States.inAsymp φ, ()⟩ := by
crPartF (FieldOp.inAsymp φ) = ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩ := by
simp [crPartF]
@[simp]
lemma crPartF_position (φ : 𝓕.PositionStates) :
crPartF (States.position φ) =
ofCrAnOpF ⟨States.position φ, CreateAnnihilate.create⟩ := by
lemma crPartF_position (φ : 𝓕.PositionFieldOp) :
crPartF (FieldOp.position φ) =
ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
simp [crPartF]
@[simp]
lemma crPartF_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
crPartF (States.outAsymp φ) = 0 := by
crPartF (FieldOp.outAsymp φ) = 0 := by
simp [crPartF]
/-- The algebra map taking an element of the free-state algbra to
the part of it in the creation and annihilation free algebra
spanned by annihilation operators. -/
def anPartF : 𝓕.States → 𝓕.FieldOpFreeAlgebra := fun φ =>
def anPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
match φ with
| States.inAsymp _ => 0
| States.position φ => ofCrAnOpF ⟨States.position φ, CreateAnnihilate.annihilate⟩
| States.outAsymp φ => ofCrAnOpF ⟨States.outAsymp φ, ()⟩
| FieldOp.inAsymp _ => 0
| FieldOp.position φ => ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩
| FieldOp.outAsymp φ => ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩
@[simp]
lemma anPartF_negAsymp (φ : 𝓕.IncomingAsymptotic) :
anPartF (States.inAsymp φ) = 0 := by
anPartF (FieldOp.inAsymp φ) = 0 := by
simp [anPartF]
@[simp]
lemma anPartF_position (φ : 𝓕.PositionStates) :
anPartF (States.position φ) =
ofCrAnOpF ⟨States.position φ, CreateAnnihilate.annihilate⟩ := by
lemma anPartF_position (φ : 𝓕.PositionFieldOp) :
anPartF (FieldOp.position φ) =
ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
simp [anPartF]
@[simp]
lemma anPartF_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
anPartF (States.outAsymp φ) = ofCrAnOpF ⟨States.outAsymp φ, ()⟩ := by
anPartF (FieldOp.outAsymp φ) = ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩ := by
simp [anPartF]
lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.States) :
lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.FieldOp) :
ofFieldOpF φ = crPartF φ + anPartF φ := by
rw [ofFieldOpF]
cases φ with
| inAsymp φ => simp [statesToCrAnType]
| position φ => simp [statesToCrAnType, CreateAnnihilate.sum_eq]
| outAsymp φ => simp [statesToCrAnType]
| inAsymp φ => simp [fieldOpToCrAnType]
| position φ => simp [fieldOpToCrAnType, CreateAnnihilate.sum_eq]
| outAsymp φ => simp [fieldOpToCrAnType]
/-!
@ -174,12 +174,12 @@ lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.States) :
-/
/-- The basis of the free creation and annihilation algebra formed by lists of CrAnStates. -/
noncomputable def ofCrAnListFBasis : Basis (List 𝓕.CrAnStates) ℂ (FieldOpFreeAlgebra 𝓕) where
/-- The basis of the free creation and annihilation algebra formed by lists of CrAnFieldOp. -/
noncomputable def ofCrAnListFBasis : Basis (List 𝓕.CrAnFieldOp) ℂ (FieldOpFreeAlgebra 𝓕) where
repr := FreeAlgebra.equivMonoidAlgebraFreeMonoid.toLinearEquiv
@[simp]
lemma ofListBasis_eq_ofList (φs : List 𝓕.CrAnStates) :
lemma ofListBasis_eq_ofList (φs : List 𝓕.CrAnFieldOp) :
ofCrAnListFBasis φs = ofCrAnListF φs := by
simp only [ofCrAnListFBasis, FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
Basis.coe_ofRepr, AlgEquiv.toLinearEquiv_symm, AlgEquiv.toLinearEquiv_apply,

12
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Grading.lean
View file

@ -24,12 +24,12 @@ noncomputable section
def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.FieldOpFreeAlgebra :=
Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ (𝓕 |>ₛ φs) = f}
lemma ofCrAnListF_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
lemma ofCrAnListF_mem_statisticSubmodule_of (φs : List 𝓕.CrAnFieldOp) (f : FieldStatistic)
(h : (𝓕 |>ₛ φs) = f) :
ofCrAnListF φs ∈ statisticSubmodule f := by
refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
lemma ofCrAnListF_bosonic_or_fermionic (φs : List 𝓕.CrAnStates) :
lemma ofCrAnListF_bosonic_or_fermionic (φs : List 𝓕.CrAnFieldOp) :
ofCrAnListF φs ∈ statisticSubmodule bosonic ∨ ofCrAnListF φs ∈ statisticSubmodule fermionic := by
by_cases h : (𝓕 |>ₛ φs) = bosonic
· left
@ -37,7 +37,7 @@ lemma ofCrAnListF_bosonic_or_fermionic (φs : List 𝓕.CrAnStates) :
· right
exact ofCrAnListF_mem_statisticSubmodule_of φs fermionic (by simpa using h)
lemma ofCrAnOpF_bosonic_or_fermionic (φ : 𝓕.CrAnStates) :
lemma ofCrAnOpF_bosonic_or_fermionic (φ : 𝓕.CrAnFieldOp) :
ofCrAnOpF φ ∈ statisticSubmodule bosonic ∨ ofCrAnOpF φ ∈ statisticSubmodule fermionic := by
rw [← ofCrAnListF_singleton]
exact ofCrAnListF_bosonic_or_fermionic [φ]
@ -50,7 +50,7 @@ def bosonicProj : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 :
else
0
lemma bosonicProj_ofCrAnListF (φs : List 𝓕.CrAnStates) :
lemma bosonicProj_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
bosonicProj (ofCrAnListF φs) = if h : (𝓕 |>ₛ φs) = bosonic then
⟨ofCrAnListF φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩ else 0 := by
conv_lhs =>
@ -110,13 +110,13 @@ def fermionicProj : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕
else
0
lemma fermionicProj_ofCrAnListF (φs : List 𝓕.CrAnStates) :
lemma fermionicProj_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
fermionicProj (ofCrAnListF φs) = if h : (𝓕 |>ₛ φs) = fermionic then
⟨ofCrAnListF φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩ else 0 := by
conv_lhs =>
rw [← ofListBasis_eq_ofList, fermionicProj, Basis.constr_basis]
lemma fermionicProj_ofCrAnListF_if_bosonic (φs : List 𝓕.CrAnStates) :
lemma fermionicProj_ofCrAnListF_if_bosonic (φs : List 𝓕.CrAnFieldOp) :
fermionicProj (ofCrAnListF φs) = if h : (𝓕 |>ₛ φs) = bosonic then
0 else ⟨ofCrAnListF φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl,
by simpa using h⟩⟩⟩ := by

2
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormTimeOrder.lean
View file

@ -35,7 +35,7 @@ def normTimeOrder : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕
@[inherit_doc normTimeOrder]
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓣𝓝ᶠ(" a ")" => normTimeOrder a
lemma normTimeOrder_ofCrAnListF (φs : List 𝓕.CrAnStates) :
lemma normTimeOrder_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
𝓣𝓝ᶠ(ofCrAnListF φs) = normTimeOrderSign φs • ofCrAnListF (normTimeOrderList φs) := by
rw [← ofListBasis_eq_ofList]
simp only [normTimeOrder, Basis.constr_basis]

92
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormalOrder.lean
View file

@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Koszul.KoszulSign
In the module
`HepLean.PerturbationTheory.FieldSpecification.NormalOrder`
we defined the normal ordering of a list of `CrAnStates`.
we defined the normal ordering of a list of `CrAnFieldOp`.
In this module we extend the normal ordering to a linear map on `FieldOpFreeAlgebra`.
We derive properties of this normal ordering.
@ -29,7 +29,7 @@ noncomputable section
/-- The linear map on the free creation and annihlation
algebra defined as the map taking
a list of CrAnStates to the normal-ordered list of states multiplied by
a list of CrAnFieldOp to the normal-ordered list of states multiplied by
the sign corresponding to the number of fermionic-fermionic
exchanges done in ordering. -/
def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
@ -39,11 +39,11 @@ def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :
@[inherit_doc normalOrderF]
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓝ᶠ(" a ")" => normalOrderF a
lemma normalOrderF_ofCrAnListF (φs : List 𝓕.CrAnStates) :
lemma normalOrderF_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF φs) = normalOrderSign φs • ofCrAnListF (normalOrderList φs) := by
rw [← ofListBasis_eq_ofList, normalOrderF, Basis.constr_basis]
lemma ofCrAnListF_eq_normalOrderF (φs : List 𝓕.CrAnStates) :
lemma ofCrAnListF_eq_normalOrderF (φs : List 𝓕.CrAnFieldOp) :
ofCrAnListF (normalOrderList φs) = normalOrderSign φs • 𝓝ᶠ(ofCrAnListF φs) := by
rw [normalOrderF_ofCrAnListF, normalOrderList, smul_smul, normalOrderSign,
Wick.koszulSign_mul_self, one_smul]
@ -119,14 +119,14 @@ lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓝ᶠ(a
-/
lemma normalOrderF_ofCrAnListF_cons_create (φ : 𝓕.CrAnStates)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (φs : List 𝓕.CrAnStates) :
lemma normalOrderF_ofCrAnListF_cons_create (φ : 𝓕.CrAnFieldOp)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (φs : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF (φ :: φs)) = ofCrAnOpF φ * 𝓝ᶠ(ofCrAnListF φs) := by
rw [normalOrderF_ofCrAnListF, normalOrderSign_cons_create φ hφ,
normalOrderList_cons_create φ hφ φs]
rw [ofCrAnListF_cons, normalOrderF_ofCrAnListF, mul_smul_comm]
lemma normalOrderF_create_mul (φ : 𝓕.CrAnStates)
lemma normalOrderF_create_mul (φ : 𝓕.CrAnFieldOp)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(ofCrAnOpF φ * a) = ofCrAnOpF φ * 𝓝ᶠ(a) := by
change (normalOrderF ∘ₗ mulLinearMap (ofCrAnOpF φ)) a =
@ -136,14 +136,14 @@ lemma normalOrderF_create_mul (φ : 𝓕.CrAnStates)
LinearMap.coe_comp, Function.comp_apply]
rw [← ofCrAnListF_cons, normalOrderF_ofCrAnListF_cons_create φ hφ]
lemma normalOrderF_ofCrAnListF_append_annihilate (φ : 𝓕.CrAnStates)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) (φs : List 𝓕.CrAnStates) :
lemma normalOrderF_ofCrAnListF_append_annihilate (φ : 𝓕.CrAnFieldOp)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) (φs : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF (φs ++ [φ])) = 𝓝ᶠ(ofCrAnListF φs) * ofCrAnOpF φ := by
rw [normalOrderF_ofCrAnListF, normalOrderSign_append_annihlate φ hφ φs,
normalOrderList_append_annihilate φ hφ φs, ofCrAnListF_append, ofCrAnListF_singleton,
normalOrderF_ofCrAnListF, smul_mul_assoc]
lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnStates)
lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnFieldOp)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate)
(a : FieldOpFreeAlgebra 𝓕) : 𝓝ᶠ(a * ofCrAnOpF φ) = 𝓝ᶠ(a) * ofCrAnOpF φ := by
change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnOpF φ)) a =
@ -154,19 +154,19 @@ lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnStates)
rw [← ofCrAnListF_singleton, ← ofCrAnListF_append, ofCrAnListF_singleton,
normalOrderF_ofCrAnListF_append_annihilate φ hφ]
lemma normalOrderF_crPartF_mul (φ : 𝓕.States) (a : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_crPartF_mul (φ : 𝓕.FieldOp) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(crPartF φ * a) =
crPartF φ * 𝓝ᶠ(a) := by
match φ with
| .inAsymp φ =>
rw [crPartF]
exact normalOrderF_create_mul ⟨States.inAsymp φ, ()⟩ rfl a
exact normalOrderF_create_mul ⟨FieldOp.inAsymp φ, ()⟩ rfl a
| .position φ =>
rw [crPartF]
exact normalOrderF_create_mul _ rfl _
| .outAsymp φ => simp
lemma normalOrderF_mul_anPartF (φ : 𝓕.States) (a : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_mul_anPartF (φ : 𝓕.FieldOp) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * anPartF φ) =
𝓝ᶠ(a) * anPartF φ := by
match φ with
@ -185,9 +185,9 @@ lemma normalOrderF_mul_anPartF (φ : 𝓕.States) (a : FieldOpFreeAlgebra 𝓕)
The main result of this section is `normalOrderF_superCommuteF_annihilate_create`.
-/
lemma normalOrderF_swap_create_annihlate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_swap_create_annihlate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(φs φs' : List 𝓕.CrAnStates) :
(φs φs' : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF φs) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) := by
rw [mul_assoc, mul_assoc, ← ofCrAnListF_cons, ← ofCrAnListF_cons, ← ofCrAnListF_append]
@ -196,9 +196,9 @@ lemma normalOrderF_swap_create_annihlate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕
rw [ofCrAnListF_append, ofCrAnListF_cons, ofCrAnListF_cons]
noncomm_ring
lemma normalOrderF_swap_create_annihlate_ofCrAnListF (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_swap_create_annihlate_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(φs : List 𝓕.CrAnStates) (a : 𝓕.FieldOpFreeAlgebra) :
(φs : List 𝓕.CrAnFieldOp) (a : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa * a) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
𝓝ᶠ(ofCrAnListF φs * ofCrAnOpF φa * ofCrAnOpF φc * a) := by
change (normalOrderF ∘ₗ mulLinearMap (ofCrAnListF φs * ofCrAnOpF φc * ofCrAnOpF φa)) a =
@ -210,7 +210,7 @@ lemma normalOrderF_swap_create_annihlate_ofCrAnListF (φc φa : 𝓕.CrAnStates)
rw [normalOrderF_swap_create_annihlate_ofCrAnListF_ofCrAnListF φc φa hφc hφa]
rfl
lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * ofCrAnOpF φc * ofCrAnOpF φa * b) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
@ -225,7 +225,7 @@ lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
normalOrderF_swap_create_annihlate_ofCrAnListF φc φa hφc hφa]
rfl
lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛca * b) = 0 := by
@ -234,7 +234,7 @@ lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
normalOrderF_swap_create_annihlate φc φa hφc hφa]
simp
lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛca * b) = 0 := by
@ -243,7 +243,7 @@ lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
exact Or.inr (normalOrderF_superCommuteF_create_annihilate φc φa hφc hφa ..)
lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.States) (a b : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * (crPartF φ) * (anPartF φ') * b) =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
𝓝ᶠ(a * (anPartF φ') * (crPartF φ) * b) := by
@ -253,22 +253,22 @@ lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.States) (a b : FieldOpFre
| .position φ, .position φ' =>
simp only [crPartF_position, anPartF_position, instCommGroup.eq_1]
rw [normalOrderF_swap_create_annihlate]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
rfl; rfl
| .inAsymp φ, .outAsymp φ' =>
simp only [crPartF_negAsymp, anPartF_posAsymp, instCommGroup.eq_1]
rw [normalOrderF_swap_create_annihlate]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
rfl; rfl
| .inAsymp φ, .position φ' =>
simp only [crPartF_negAsymp, anPartF_position, instCommGroup.eq_1]
rw [normalOrderF_swap_create_annihlate]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
rfl; rfl
| .position φ, .outAsymp φ' =>
simp only [crPartF_position, anPartF_posAsymp, instCommGroup.eq_1]
rw [normalOrderF_swap_create_annihlate]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
rfl; rfl
/-!
@ -279,13 +279,13 @@ Using the results from above.
-/
lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.States) (a b : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * (anPartF φ) * (crPartF φ') * b) =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝ᶠ(a * (crPartF φ') *
(anPartF φ) * b) := by
simp [normalOrderF_swap_crPartF_anPartF, smul_smul]
lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) (a b : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * superCommuteF
(crPartF φ) (anPartF φ') * b) = 0 := by
match φ, φ' with
@ -304,7 +304,7 @@ lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) (a b : F
rw [crPartF_position, anPartF_posAsymp]
exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) (a b : FieldOpFreeAlgebra 𝓕) :
lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * superCommuteF
(anPartF φ) (crPartF φ') * b) = 0 := by
match φ, φ' with
@ -330,7 +330,7 @@ lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) (a b : F
-/
@[simp]
lemma normalOrderF_crPartF_mul_crPartF (φ φ' : 𝓕.States) :
lemma normalOrderF_crPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
𝓝ᶠ(crPartF φ * crPartF φ') =
crPartF φ * crPartF φ' := by
rw [normalOrderF_crPartF_mul]
@ -339,7 +339,7 @@ lemma normalOrderF_crPartF_mul_crPartF (φ φ' : 𝓕.States) :
simp
@[simp]
lemma normalOrderF_anPartF_mul_anPartF (φ φ' : 𝓕.States) :
lemma normalOrderF_anPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
𝓝ᶠ(anPartF φ * anPartF φ') =
anPartF φ * anPartF φ' := by
rw [normalOrderF_mul_anPartF]
@ -348,7 +348,7 @@ lemma normalOrderF_anPartF_mul_anPartF (φ φ' : 𝓕.States) :
simp
@[simp]
lemma normalOrderF_crPartF_mul_anPartF (φ φ' : 𝓕.States) :
lemma normalOrderF_crPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
𝓝ᶠ(crPartF φ * anPartF φ') =
crPartF φ * anPartF φ' := by
rw [normalOrderF_crPartF_mul]
@ -357,7 +357,7 @@ lemma normalOrderF_crPartF_mul_anPartF (φ φ' : 𝓕.States) :
simp
@[simp]
lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.States) :
lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
𝓝ᶠ(anPartF φ * crPartF φ') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
(crPartF φ' * anPartF φ) := by
@ -367,7 +367,7 @@ lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.States) :
rw [← mul_assoc, normalOrderF_swap_anPartF_crPartF]
simp
lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.States) :
lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
𝓝ᶠ(ofFieldOpF φ * ofFieldOpF φ') =
crPartF φ * crPartF φ' +
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
@ -388,8 +388,8 @@ lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.States) :
TODO "Split the following two lemmas up into smaller parts."
lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
(φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnFieldOp) :
(𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca * ofCrAnListF φs')) =
normalOrderSign (φs ++ φc' :: φc :: φs') •
(ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca *
@ -447,10 +447,10 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
rw [ofCrAnListF_append, ofCrAnListF_singleton, ofCrAnListF_singleton, smul_mul_assoc]
lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
(φa φa' : 𝓕.CrAnStates)
(φa φa' : 𝓕.CrAnFieldOp)
(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
(φs φs' : List 𝓕.CrAnStates) :
(φs φs' : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * ofCrAnListF φs') =
normalOrderSign (φs ++ φa' :: φa :: φs') •
(ofCrAnListF (createFilter (φs ++ φs'))
@ -520,15 +520,15 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
-/
lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.CrAnStates) :
lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛca =
ofCrAnListF φs * 𝓝ᶠ(ofCrAnListF φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs := by
simp [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF, ofCrAnListF_append,
smul_sub, smul_smul, mul_comm]
lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca =
lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca =
ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs := by
rw [ofFieldOpListF_sum, map_sum, Finset.mul_sum, Finset.smul_sum, Finset.sum_mul,
@ -544,21 +544,21 @@ lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrA
-/
lemma ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) :
lemma ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) :
ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs
+ [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
simp [ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF]
lemma ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.CrAnStates)
(φs' : List 𝓕.States) : ofCrAnOpF φ * 𝓝ᶠ(ofFieldOpListF φs') =
lemma ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : ofCrAnOpF φ * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnOpF φ
+ [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
simp [← ofCrAnListF_singleton, ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF]
lemma anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States)
(φs' : List 𝓕.States) :
lemma anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp)
(φs' : List 𝓕.FieldOp) :
anPartF φ * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs' * anPartF φ)
+ [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by

140
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/SuperCommute.lean
View file

@ -42,13 +42,13 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
-/
lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, ofCrAnListF φs']ₛca =
ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs) := by
rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
simp only [superCommuteF, Basis.constr_basis]
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnFieldOp) :
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
ofCrAnOpF φ * ofCrAnOpF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnOpF φ' * ofCrAnOpF φ := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
@ -57,7 +57,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnStates) :
rw [ofCrAnListF_append]
rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnFieldOp) (φs : List 𝓕.FieldOp) :
[ofCrAnListF φcas, ofFieldOpListF φs]ₛca = ofCrAnListF φcas * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnListF φcas := by
conv_lhs => rw [ofFieldOpListF_sum]
@ -70,7 +70,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (
← Finset.sum_mul, ← ofFieldOpListF_sum]
simp
lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
[ofFieldOpListF φ, ofFieldOpListF φs]ₛca = ofFieldOpListF φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpListF φ := by
conv_lhs => rw [ofFieldOpListF_sum]
@ -83,165 +83,165 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs
Algebra.smul_mul_assoc, Finset.sum_sub_distrib]
rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofFieldOpListF_sum]
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
[ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
simp
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.States) (φ : 𝓕.States) :
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
[ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
simp
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
[anPartF φ, crPartF φ']ₛca = anPartF φ * crPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
| FieldOp.inAsymp φ, _ =>
simp
| _, States.outAsymp φ =>
| _, FieldOp.outAsymp φ =>
simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
sub_self]
| States.position φ, States.position φ' =>
| FieldOp.position φ, FieldOp.position φ' =>
simp only [anPartF_position, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.outAsymp φ, States.position φ' =>
| FieldOp.outAsymp φ, FieldOp.position φ' =>
simp only [anPartF_posAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.position φ, States.inAsymp φ' =>
| FieldOp.position φ, FieldOp.inAsymp φ' =>
simp only [anPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp only [List.singleton_append, instCommGroup.eq_1, crAnStatistics,
FieldStatistic.ofList_singleton, Function.comp_apply, crAnStatesToStates_prod, ←
FieldStatistic.ofList_singleton, Function.comp_apply, crAnFieldOpToFieldOp_prod, ←
ofCrAnListF_append]
| States.outAsymp φ, States.inAsymp φ' =>
| FieldOp.outAsymp φ, FieldOp.inAsymp φ' =>
simp only [anPartF_posAsymp, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) :
[crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
| FieldOp.outAsymp φ, _ =>
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
mul_zero, sub_self]
| _, States.inAsymp φ =>
| _, FieldOp.inAsymp φ =>
simp only [anPartF_negAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
sub_self]
| States.position φ, States.position φ' =>
| FieldOp.position φ, FieldOp.position φ' =>
simp only [crPartF_position, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.position φ, States.outAsymp φ' =>
| FieldOp.position φ, FieldOp.outAsymp φ' =>
simp only [crPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.inAsymp φ, States.position φ' =>
| FieldOp.inAsymp φ, FieldOp.position φ' =>
simp only [crPartF_negAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.inAsymp φ, States.outAsymp φ' =>
| FieldOp.inAsymp φ, FieldOp.outAsymp φ' =>
simp only [crPartF_negAsymp, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.FieldOp) :
[crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := by
match φ, φ' with
| States.outAsymp φ, _ =>
| FieldOp.outAsymp φ, _ =>
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
mul_zero, sub_self]
| _, States.outAsymp φ =>
| _, FieldOp.outAsymp φ =>
simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
sub_self]
| States.position φ, States.position φ' =>
| FieldOp.position φ, FieldOp.position φ' =>
simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.position φ, States.inAsymp φ' =>
| FieldOp.position φ, FieldOp.inAsymp φ' =>
simp only [crPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.inAsymp φ, States.position φ' =>
| FieldOp.inAsymp φ, FieldOp.position φ' =>
simp only [crPartF_negAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.inAsymp φ, States.inAsymp φ' =>
| FieldOp.inAsymp φ, FieldOp.inAsymp φ' =>
simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.States) :
lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.FieldOp) :
[anPartF φ, anPartF φ']ₛca =
anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := by
match φ, φ' with
| States.inAsymp φ, _ =>
| FieldOp.inAsymp φ, _ =>
simp
| _, States.inAsymp φ =>
| _, FieldOp.inAsymp φ =>
simp
| States.position φ, States.position φ' =>
| FieldOp.position φ, FieldOp.position φ' =>
simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.position φ, States.outAsymp φ' =>
| FieldOp.position φ, FieldOp.outAsymp φ' =>
simp only [anPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.outAsymp φ, States.position φ' =>
| FieldOp.outAsymp φ, FieldOp.position φ' =>
simp only [anPartF_posAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
| States.outAsymp φ, States.outAsymp φ' =>
| FieldOp.outAsymp φ, FieldOp.outAsymp φ' =>
simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
simp [crAnStatistics, ← ofCrAnListF_append]
lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
[crPartF φ, ofFieldOpListF φs]ₛca =
crPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs *
crPartF φ := by
match φ with
| States.inAsymp φ =>
| FieldOp.inAsymp φ =>
simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
simp [crAnStatistics]
| States.position φ =>
| FieldOp.position φ =>
simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
simp [crAnStatistics]
| States.outAsymp φ =>
| FieldOp.outAsymp φ =>
simp
lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
[anPartF φ, ofFieldOpListF φs]ₛca =
anPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
ofFieldOpListF φs * anPartF φ := by
match φ with
| States.inAsymp φ =>
| FieldOp.inAsymp φ =>
simp
| States.position φ =>
| FieldOp.position φ =>
simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
simp [crAnStatistics]
| States.outAsymp φ =>
| FieldOp.outAsymp φ =>
simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
simp [crAnStatistics]
lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.States) :
lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
[crPartF φ, ofFieldOpF φ']ₛca =
crPartF φ * ofFieldOpF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
simp
lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.States) :
lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
[anPartF φ, ofFieldOpF φ']ₛca =
anPartF φ * ofFieldOpF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
@ -256,44 +256,44 @@ Lemmas which rewrite a multiplication of two elements of the algebra as their co
multiplication with a sign plus the super commutor.
-/
lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnStates) :
lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
ofCrAnListF φs * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnListF φs
+ [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
simp [ofCrAnListF_append]
lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.CrAnFieldOp) :
ofCrAnOpF φ * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnOpF φ
+ [ofCrAnOpF φ, ofCrAnListF φs']ₛca := by
rw [← ofCrAnListF_singleton, ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF]
simp
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.States) :
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.FieldOp) :
ofFieldOpListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
simp
lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
+ [ofFieldOpF φ, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
simp
lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofFieldOpF φ]ₛca := by
rw [superCommuteF_ofFieldOpListF_ofFieldOpF]
simp
lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
crPartF φ * anPartF φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ +
[crPartF φ, anPartF φ']ₛca := by
rw [superCommuteF_crPartF_anPartF]
simp
lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
anPartF φ * crPartF φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
crPartF φ' * anPartF φ +
@ -301,20 +301,20 @@ lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
rw [superCommuteF_anPartF_crPartF]
simp
lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
crPartF φ * crPartF φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ +
[crPartF φ, crPartF φ']ₛca := by
rw [superCommuteF_crPartF_crPartF]
simp
lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
anPartF φ * anPartF φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ +
[anPartF φ, anPartF φ']ₛca := by
rw [superCommuteF_anPartF_anPartF]
simp
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp) (φs' : List 𝓕.FieldOp) :
ofCrAnListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
+ [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
@ -326,7 +326,7 @@ lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnState
-/
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, ofCrAnListF φs']ₛca =
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛca := by
rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF, smul_sub]
@ -338,7 +338,7 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnState
simp only [one_smul]
abel
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛca := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
@ -357,7 +357,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnStates) :
-/
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛca =
[ofCrAnListF φs, ofCrAnOpF φ]ₛca * ofCrAnListF φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
@ -377,8 +377,8 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs φs
rw [← ofCrAnListF_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
simp only [instCommGroup, map_mul, mul_comm]
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
[ofCrAnListF φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
@ -402,8 +402,8 @@ Within the creation and annihilation algebra, we have that
`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
-/
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnStates) :
(φs' : List 𝓕.CrAnStates) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
(φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛca *
ofCrAnListF (φs'.drop (n + 1))
@ -417,7 +417,7 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnStates)
· simp [Finset.mul_sum, smul_smul, ofCrAnListF_cons, mul_assoc,
FieldStatistic.ofList_cons_eq_mul, mul_comm]
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) : (φs' : List 𝓕.FieldOp) →
[ofCrAnListF φs, ofFieldOpListF φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛca *
@ -436,7 +436,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnState
· simp [Finset.mul_sum, smul_smul, ofFieldOpListF_cons, mul_assoc,
FieldStatistic.ofList_cons_eq_mul, mul_comm]
lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛca]ₛca =
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛca]ₛca -
@ -706,7 +706,7 @@ lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.FieldOpFreeAlge
superCommuteF_fermionic_fermionic (by simp) (by simp)]
abel
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic := by
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
@ -743,13 +743,13 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : Lis
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnFieldOp) :
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule fermionic := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
exact superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [φ']
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ2 φ3 with hs | hs
@ -779,7 +779,7 @@ lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 :
rw [h]
apply superCommuteF_grade h1 hs
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
(ha : a ∈ statisticSubmodule bosonic) :
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
@ -808,7 +808,7 @@ lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φ
simp_all [p, Finset.smul_sum]
· exact ha
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
(ha : a ∈ statisticSubmodule fermionic) :
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
@ -842,7 +842,7 @@ lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (
simp [smul_smul, mul_comm]
· exact ha
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnFieldOp}
(h : [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic) :
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0 := by
by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0

34
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/TimeOrder.lean
View file

@ -34,7 +34,7 @@ def timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
@[inherit_doc timeOrderF]
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓣ᶠ(" a ")" => timeOrderF a
lemma timeOrderF_ofCrAnListF (φs : List 𝓕.CrAnStates) :
lemma timeOrderF_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
𝓣ᶠ(ofCrAnListF φs) = crAnTimeOrderSign φs • ofCrAnListF (crAnTimeOrderList φs) := by
rw [← ofListBasis_eq_ofList]
simp only [timeOrderF, Basis.constr_basis]
@ -99,7 +99,7 @@ lemma timeOrderF_timeOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b
· rw [timeOrderF_timeOrderF_mid]
simp
lemma timeOrderF_ofFieldOpListF (φs : List 𝓕.States) :
lemma timeOrderF_ofFieldOpListF (φs : List 𝓕.FieldOp) :
𝓣ᶠ(ofFieldOpListF φs) = timeOrderSign φs • ofFieldOpListF (timeOrderList φs) := by
conv_lhs =>
rw [ofFieldOpListF_sum, map_sum]
@ -116,10 +116,10 @@ lemma timeOrderF_ofFieldOpListF_nil : timeOrderF (𝓕 := 𝓕) (ofFieldOpListF
simp [timeOrderSign, Wick.koszulSign, timeOrderList]
@[simp]
lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.FieldOp) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
simp [timeOrderF_ofFieldOpListF, timeOrderSign, timeOrderList]
lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.FieldOp} (h : timeOrderRel φ ψ) :
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = ofFieldOpF φ * ofFieldOpF ψ := by
rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton, ← ofFieldOpListF_append,
timeOrderF_ofFieldOpListF]
@ -127,7 +127,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOr
rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
simp
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.FieldOp} (h : ¬ timeOrderRel φ ψ) :
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOpF ψ * ofFieldOpF φ := by
rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton,
← ofFieldOpListF_append, timeOrderF_ofFieldOpListF]
@ -135,7 +135,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬
rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
simp [← ofFieldOpListF_append]
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.FieldOp}
(h : ¬ timeOrderRel φ ψ) :
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofFieldOpF ψ * ofFieldOpF φ) := by
rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
@ -145,7 +145,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.S
simp_all
lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) :
𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
@ -163,28 +163,28 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
simp_all
lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_right
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
rw [timeOrderF_timeOrderF_right,
timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_left
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * a) = 0 := by
rw [timeOrderF_timeOrderF_left,
timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_mid
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
{φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
rw [timeOrderF_timeOrderF_mid,
timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
{φ1 φ2 : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ([a, [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca]ₛca) = 0 := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
@ -207,7 +207,7 @@ lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
simp
lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
{φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
@ -223,7 +223,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
simp
lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel'
{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
{φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
@ -239,7 +239,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel'
simp
lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
(φ1 φ2 φ3 : 𝓕.CrAnFieldOp) (h : ¬
(crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2)) :
@ -277,7 +277,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
exact IsTrans.trans φ3 φ2 φ1 h32 h21
lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_eq_time
{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
{φ ψ : 𝓕.CrAnFieldOp} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
@ -297,7 +297,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_eq_time
/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`
where `φᵢ` is the state
which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`. -/
lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
𝓣ᶠ(ofFieldOpListF (φ :: φs)) =
𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
@ -312,7 +312,7 @@ lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States)
where `φᵢ` is the state
which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`.
Here `s` is written using finite sets. -/
lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
𝓣ᶠ(ofFieldOpListF (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
(Finset.filter (fun x =>
(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •