refactor: Rename CrAnAlgebra
This commit is contained in:
jstoobysmith
parent
9a5676e134
commit
b0735a1e13
16 changed files with 214 additions and 214 deletions
|
|
@ -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.Algebras.CrAnAlgebra.SuperCommute
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.SuperCommute
|
||||
import Mathlib.Algebra.RingQuot
|
||||
import Mathlib.RingTheory.TwoSidedIdeal.Operations
|
||||
/-!
|
||||
|
|
@ -13,7 +13,7 @@ import Mathlib.RingTheory.TwoSidedIdeal.Operations
|
|||
-/
|
||||
|
||||
namespace FieldSpecification
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
open HepLean.List
|
||||
open FieldStatistic
|
||||
|
||||
|
|
@ -21,7 +21,7 @@ variable (𝓕 : FieldSpecification)
|
|||
|
||||
/-- The set contains the super-commutors equal to zero in the operator algebra.
|
||||
This contains e.g. the super-commutor of two creation operators. -/
|
||||
def fieldOpIdealSet : Set (CrAnAlgebra 𝓕) :=
|
||||
def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
|
||||
{ x |
|
||||
(∃ (φ1 φ2 φ3 : 𝓕.CrAnStates),
|
||||
x = [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca)
|
||||
|
|
@ -39,16 +39,16 @@ abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCo
|
|||
namespace FieldOpAlgebra
|
||||
variable {𝓕 : FieldSpecification}
|
||||
|
||||
/-- The instance of a setoid on `CrAnAlgebra` from the ideal `TwoSidedIdeal`. -/
|
||||
instance : Setoid (CrAnAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
|
||||
/-- The instance of a setoid on `FieldOpFreeAlgebra` from the ideal `TwoSidedIdeal`. -/
|
||||
instance : Setoid (FieldOpFreeAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
|
||||
|
||||
lemma equiv_iff_sub_mem_ideal (x y : CrAnAlgebra 𝓕) :
|
||||
lemma equiv_iff_sub_mem_ideal (x y : FieldOpFreeAlgebra 𝓕) :
|
||||
x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||||
rw [← TwoSidedIdeal.rel_iff]
|
||||
rfl
|
||||
|
||||
/-- The projection of `CrAnAlgebra` down to `FieldOpAlgebra` as an algebra map. -/
|
||||
def ι : CrAnAlgebra 𝓕 →ₐ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
/-- The projection of `FieldOpFreeAlgebra` down to `FieldOpAlgebra` as an algebra map. -/
|
||||
def ι : FieldOpFreeAlgebra 𝓕 →ₐ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.mk'
|
||||
map_one' := by rfl
|
||||
map_mul' x y := by rfl
|
||||
|
|
@ -62,9 +62,9 @@ lemma ι_surjective : Function.Surjective (@ι 𝓕) := by
|
|||
use x
|
||||
rfl
|
||||
|
||||
lemma ι_apply (x : CrAnAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
|
||||
lemma ι_apply (x : FieldOpFreeAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
|
||||
|
||||
lemma ι_of_mem_fieldOpIdealSet (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
ι x = 0 := by
|
||||
rw [ι_apply]
|
||||
change ⟦x⟧ = ⟦0⟧
|
||||
|
|
@ -157,8 +157,8 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2
|
|||
simp [ofCrAnList_singleton, ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnState]
|
||||
|
||||
@[simp]
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
|
||||
(a : 𝓕.CrAnAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
|
||||
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnStates)
|
||||
(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
|
||||
change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
|
||||
have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
|
||||
apply (ofCrAnListBasis.ext fun l ↦ ?_)
|
||||
|
|
@ -166,8 +166,8 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ
|
|||
rw [h1]
|
||||
simp
|
||||
|
||||
lemma ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
|
||||
(a : 𝓕.CrAnAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
|
||||
lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
|
||||
(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
|
||||
ι [ofCrAnState φ1, ofCrAnState φ2]ₛca * ι a = 0 := by
|
||||
rcases ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
|
||||
swap
|
||||
|
|
@ -182,7 +182,7 @@ lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStat
|
|||
rw [Subalgebra.mem_center_iff]
|
||||
intro a
|
||||
obtain ⟨a, rfl⟩ := ι_surjective a
|
||||
have h0 := ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
|
||||
have h0 := ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
|
||||
trans ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
|
||||
swap
|
||||
simp only [add_zero]
|
||||
|
|
@ -194,7 +194,7 @@ lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStat
|
|||
## The kernal of ι
|
||||
-/
|
||||
|
||||
lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
|
||||
lemma ι_eq_zero_iff_mem_ideal (x : FieldOpFreeAlgebra 𝓕) :
|
||||
ι x = 0 ↔ x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||||
rw [ι_apply]
|
||||
change ⟦x⟧ = ⟦0⟧ ↔ _
|
||||
|
|
@ -203,7 +203,7 @@ lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
|
|||
simp only
|
||||
rfl
|
||||
|
||||
lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
|
||||
have hx' := hx
|
||||
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
|
||||
|
|
@ -234,7 +234,7 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
|
|||
· right
|
||||
rw [bosonicProj_of_mem_fermionic _ h]
|
||||
|
||||
lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||||
x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
|
||||
have hx' := hx
|
||||
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
|
||||
|
|
@ -265,10 +265,10 @@ lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x
|
|||
rw [fermionicProj_of_mem_fermionic _ h]
|
||||
simpa using hx'
|
||||
|
||||
lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||||
lemma bosonicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||||
x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
|
||||
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
|
||||
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
|
||||
a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
|
||||
change p x hx
|
||||
apply AddSubgroup.closure_induction
|
||||
|
|
@ -401,7 +401,7 @@ lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.spa
|
|||
· intro x hx
|
||||
simp [p]
|
||||
|
||||
lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||||
lemma fermionicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||||
x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||||
have hb := bosonicProj_mem_ideal x hx
|
||||
rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢
|
||||
|
|
@ -409,7 +409,7 @@ lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.s
|
|||
simp only [map_add] at hx
|
||||
simp_all
|
||||
|
||||
lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
|
||||
lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : FieldOpFreeAlgebra 𝓕) :
|
||||
ι x = 0 ↔ ι x.bosonicProj.1 = 0 ∧ ι x.fermionicProj.1 = 0 := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue