refactor: Rename States to FieldOps
This commit is contained in:
jstoobysmith
parent
171e80fc04
commit
8f41de5785
36 changed files with 946 additions and 946 deletions
|
|
@ -23,13 +23,13 @@ variable (𝓕 : FieldSpecification)
|
|||
This contains e.g. the super-commutor of two creation operators. -/
|
||||
def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
|
||||
{ x |
|
||||
(∃ (φ1 φ2 φ3 : 𝓕.CrAnStates),
|
||||
(∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
|
||||
x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca)
|
||||
∨ (∃ (φc φc' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
|
||||
∨ (∃ (φc φc' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
|
||||
x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛca)
|
||||
∨ (∃ (φa φa' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
|
||||
∨ (∃ (φa φa' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
|
||||
x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛca)
|
||||
∨ (∃ (φ φ' : 𝓕.CrAnStates) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
|
||||
∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
|
||||
x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
|
||||
|
||||
/-- The algebra spanned by cr and an parts of fields, with appropriate super-commutors
|
||||
|
|
@ -72,7 +72,7 @@ lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.f
|
|||
refine RingConGen.Rel.of x 0 ?_
|
||||
simpa using hx
|
||||
|
||||
lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = .create)
|
||||
lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = .create)
|
||||
(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛca = 0 := by
|
||||
apply ι_of_mem_fieldOpIdealSet
|
||||
simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
|
||||
|
|
@ -81,7 +81,7 @@ lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc :
|
|||
left
|
||||
use φc, φc', hφc, hφc'
|
||||
|
||||
lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
|
||||
lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
|
||||
(hφa : 𝓕 |>ᶜ φa = .annihilate) (hφa' : 𝓕 |>ᶜ φa' = .annihilate) :
|
||||
ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛca = 0 := by
|
||||
apply ι_of_mem_fieldOpIdealSet
|
||||
|
|
@ -92,7 +92,7 @@ lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
|
|||
left
|
||||
use φa, φa', hφa, hφa'
|
||||
|
||||
lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
|
||||
lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnFieldOp}
|
||||
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
|
||||
apply ι_of_mem_fieldOpIdealSet
|
||||
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
|
||||
|
|
@ -101,7 +101,7 @@ lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
|
|||
right
|
||||
use φ, ψ
|
||||
|
||||
lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
|
||||
lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnFieldOp)
|
||||
(h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule fermionic) :
|
||||
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton] at h ⊢
|
||||
|
|
@ -111,7 +111,7 @@ lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
|
|||
simpa using h
|
||||
· simp [h]
|
||||
|
||||
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
|
||||
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule bosonic ∨
|
||||
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
|
||||
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [ψ] with h | h
|
||||
|
|
@ -127,14 +127,14 @@ lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnSta
|
|||
-/
|
||||
|
||||
@[simp]
|
||||
lemma ι_superCommuteF_ofCrAnOpF_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnStates) :
|
||||
lemma ι_superCommuteF_ofCrAnOpF_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
|
||||
ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca = 0 := by
|
||||
apply ι_of_mem_fieldOpIdealSet
|
||||
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
|
||||
left
|
||||
use φ1, φ2, φ3
|
||||
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnStates) :
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
|
||||
ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnOpF φ3]ₛca = 0 := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||||
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
|
||||
|
|
@ -146,8 +146,8 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3
|
|||
· rw [superCommuteF_fermionic_fermionic_symm h h']
|
||||
simp [ofCrAnListF_singleton]
|
||||
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 : 𝓕.CrAnStates)
|
||||
(φs : List 𝓕.CrAnStates) :
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 : 𝓕.CrAnFieldOp)
|
||||
(φs : List 𝓕.CrAnFieldOp) :
|
||||
ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnListF φs]ₛca = 0 := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||||
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
|
||||
|
|
@ -157,7 +157,7 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 :
|
|||
simp [ofCrAnListF_singleton, ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF]
|
||||
|
||||
@[simp]
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnStates)
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnFieldOp)
|
||||
(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca = 0 := by
|
||||
change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) a = _
|
||||
have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) = 0 := by
|
||||
|
|
@ -166,7 +166,7 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1
|
|||
rw [h1]
|
||||
simp
|
||||
|
||||
lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 : 𝓕.CrAnStates)
|
||||
lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 : 𝓕.CrAnFieldOp)
|
||||
(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca -
|
||||
ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca * ι a = 0 := by
|
||||
rcases ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero φ1 φ2 with h | h
|
||||
|
|
@ -177,7 +177,7 @@ lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 :
|
|||
simp
|
||||
· simp
|
||||
|
||||
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnStates) :
|
||||
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnFieldOp) :
|
||||
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
|
||||
rw [Subalgebra.mem_center_iff]
|
||||
intro a
|
||||
|
|
@ -428,111 +428,111 @@ lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : FieldOpFreeAlgebra
|
|||
|
||||
-/
|
||||
|
||||
/-- An element of `FieldOpAlgebra` from a `States`. -/
|
||||
def ofFieldOp (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
|
||||
/-- An element of `FieldOpAlgebra` from a `FieldOp`. -/
|
||||
def ofFieldOp (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
|
||||
|
||||
lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.States) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
|
||||
lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.FieldOp) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
|
||||
|
||||
/-- An element of `FieldOpAlgebra` from a list of `States`. -/
|
||||
def ofFieldOpList (φs : List 𝓕.States) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
|
||||
/-- An element of `FieldOpAlgebra` from a list of `FieldOp`. -/
|
||||
def ofFieldOpList (φs : List 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
|
||||
|
||||
lemma ofFieldOpList_eq_ι_ofFieldOpListF (φs : List 𝓕.States) :
|
||||
lemma ofFieldOpList_eq_ι_ofFieldOpListF (φs : List 𝓕.FieldOp) :
|
||||
ofFieldOpList φs = ι (ofFieldOpListF φs) := rfl
|
||||
|
||||
lemma ofFieldOpList_append (φs ψs : List 𝓕.States) :
|
||||
lemma ofFieldOpList_append (φs ψs : List 𝓕.FieldOp) :
|
||||
ofFieldOpList (φs ++ ψs) = ofFieldOpList φs * ofFieldOpList ψs := by
|
||||
simp only [ofFieldOpList]
|
||||
rw [ofFieldOpListF_append]
|
||||
simp
|
||||
|
||||
lemma ofFieldOpList_cons (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma ofFieldOpList_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
|
||||
ofFieldOpList (φ :: φs) = ofFieldOp φ * ofFieldOpList φs := by
|
||||
simp only [ofFieldOpList]
|
||||
rw [ofFieldOpListF_cons]
|
||||
simp only [map_mul]
|
||||
rfl
|
||||
|
||||
lemma ofFieldOpList_singleton (φ : 𝓕.States) :
|
||||
lemma ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
|
||||
ofFieldOpList [φ] = ofFieldOp φ := by
|
||||
simp only [ofFieldOpList, ofFieldOp, ofFieldOpListF_singleton]
|
||||
|
||||
/-- An element of `FieldOpAlgebra` from a `CrAnStates`. -/
|
||||
def ofCrAnFieldOp (φ : 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
|
||||
/-- An element of `FieldOpAlgebra` from a `CrAnFieldOp`. -/
|
||||
def ofCrAnFieldOp (φ : 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
|
||||
|
||||
lemma ofCrAnFieldOp_eq_ι_ofCrAnOpF (φ : 𝓕.CrAnStates) :
|
||||
lemma ofCrAnFieldOp_eq_ι_ofCrAnOpF (φ : 𝓕.CrAnFieldOp) :
|
||||
ofCrAnFieldOp φ = ι (ofCrAnOpF φ) := rfl
|
||||
|
||||
lemma ofFieldOp_eq_sum (φ : 𝓕.States) :
|
||||
ofFieldOp φ = (∑ i : 𝓕.statesToCrAnType φ, ofCrAnFieldOp ⟨φ, i⟩) := by
|
||||
lemma ofFieldOp_eq_sum (φ : 𝓕.FieldOp) :
|
||||
ofFieldOp φ = (∑ i : 𝓕.fieldOpToCrAnType φ, ofCrAnFieldOp ⟨φ, i⟩) := by
|
||||
rw [ofFieldOp, ofFieldOpF]
|
||||
simp only [map_sum]
|
||||
rfl
|
||||
|
||||
/-- An element of `FieldOpAlgebra` from a list of `CrAnStates`. -/
|
||||
def ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
|
||||
/-- An element of `FieldOpAlgebra` from a list of `CrAnFieldOp`. -/
|
||||
def ofCrAnFieldOpList (φs : List 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
|
||||
|
||||
lemma ofCrAnFieldOpList_eq_ι_ofCrAnListF (φs : List 𝓕.CrAnStates) :
|
||||
lemma ofCrAnFieldOpList_eq_ι_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
|
||||
ofCrAnFieldOpList φs = ι (ofCrAnListF φs) := rfl
|
||||
|
||||
lemma ofCrAnFieldOpList_append (φs ψs : List 𝓕.CrAnStates) :
|
||||
lemma ofCrAnFieldOpList_append (φs ψs : List 𝓕.CrAnFieldOp) :
|
||||
ofCrAnFieldOpList (φs ++ ψs) = ofCrAnFieldOpList φs * ofCrAnFieldOpList ψs := by
|
||||
simp only [ofCrAnFieldOpList]
|
||||
rw [ofCrAnListF_append]
|
||||
simp
|
||||
|
||||
lemma ofCrAnFieldOpList_singleton (φ : 𝓕.CrAnStates) :
|
||||
lemma ofCrAnFieldOpList_singleton (φ : 𝓕.CrAnFieldOp) :
|
||||
ofCrAnFieldOpList [φ] = ofCrAnFieldOp φ := by
|
||||
simp only [ofCrAnFieldOpList, ofCrAnFieldOp, ofCrAnListF_singleton]
|
||||
|
||||
lemma ofFieldOpList_eq_sum (φs : List 𝓕.States) :
|
||||
lemma ofFieldOpList_eq_sum (φs : List 𝓕.FieldOp) :
|
||||
ofFieldOpList φs = ∑ s : CrAnSection φs, ofCrAnFieldOpList s.1 := by
|
||||
rw [ofFieldOpList, ofFieldOpListF_sum]
|
||||
simp only [map_sum]
|
||||
rfl
|
||||
|
||||
/-- The annihilation part of a state. -/
|
||||
def anPart (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
|
||||
def anPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
|
||||
|
||||
lemma anPart_eq_ι_anPartF (φ : 𝓕.States) : anPart φ = ι (anPartF φ) := rfl
|
||||
lemma anPart_eq_ι_anPartF (φ : 𝓕.FieldOp) : anPart φ = ι (anPartF φ) := rfl
|
||||
|
||||
@[simp]
|
||||
lemma anPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
|
||||
anPart (States.inAsymp φ) = 0 := by
|
||||
anPart (FieldOp.inAsymp φ) = 0 := by
|
||||
simp [anPart, anPartF]
|
||||
|
||||
@[simp]
|
||||
lemma anPart_position (φ : 𝓕.PositionStates) :
|
||||
anPart (States.position φ) =
|
||||
ofCrAnFieldOp ⟨States.position φ, CreateAnnihilate.annihilate⟩ := by
|
||||
lemma anPart_position (φ : 𝓕.PositionFieldOp) :
|
||||
anPart (FieldOp.position φ) =
|
||||
ofCrAnFieldOp ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
|
||||
simp [anPart, ofCrAnFieldOp]
|
||||
|
||||
@[simp]
|
||||
lemma anPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
|
||||
anPart (States.outAsymp φ) = ofCrAnFieldOp ⟨States.outAsymp φ, ()⟩ := by
|
||||
anPart (FieldOp.outAsymp φ) = ofCrAnFieldOp ⟨FieldOp.outAsymp φ, ()⟩ := by
|
||||
simp [anPart, ofCrAnFieldOp]
|
||||
|
||||
/-- The creation part of a state. -/
|
||||
def crPart (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
|
||||
def crPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
|
||||
|
||||
lemma crPart_eq_ι_crPartF (φ : 𝓕.States) : crPart φ = ι (crPartF φ) := rfl
|
||||
lemma crPart_eq_ι_crPartF (φ : 𝓕.FieldOp) : crPart φ = ι (crPartF φ) := rfl
|
||||
|
||||
@[simp]
|
||||
lemma crPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
|
||||
crPart (States.inAsymp φ) = ofCrAnFieldOp ⟨States.inAsymp φ, ()⟩ := by
|
||||
crPart (FieldOp.inAsymp φ) = ofCrAnFieldOp ⟨FieldOp.inAsymp φ, ()⟩ := by
|
||||
simp [crPart, ofCrAnFieldOp]
|
||||
|
||||
@[simp]
|
||||
lemma crPart_position (φ : 𝓕.PositionStates) :
|
||||
crPart (States.position φ) =
|
||||
ofCrAnFieldOp ⟨States.position φ, CreateAnnihilate.create⟩ := by
|
||||
lemma crPart_position (φ : 𝓕.PositionFieldOp) :
|
||||
crPart (FieldOp.position φ) =
|
||||
ofCrAnFieldOp ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
|
||||
simp [crPart, ofCrAnFieldOp]
|
||||
|
||||
@[simp]
|
||||
lemma crPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
|
||||
crPart (States.outAsymp φ) = 0 := by
|
||||
crPart (FieldOp.outAsymp φ) = 0 := by
|
||||
simp [crPart]
|
||||
|
||||
lemma ofFieldOp_eq_crPart_add_anPart (φ : 𝓕.States) :
|
||||
lemma ofFieldOp_eq_crPart_add_anPart (φ : 𝓕.FieldOp) :
|
||||
ofFieldOp φ = crPart φ + anPart φ := by
|
||||
rw [ofFieldOp, crPart, anPart, ofFieldOpF_eq_crPartF_add_anPartF]
|
||||
simp only [map_add]
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue