600 lines
24 KiB
Text
600 lines
24 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.FieldOpFreeAlgebra.SuperCommute
|
||
import Mathlib.Algebra.RingQuot
|
||
import Mathlib.RingTheory.TwoSidedIdeal.Operations
|
||
/-!
|
||
|
||
# Field operator algebra
|
||
|
||
-/
|
||
|
||
namespace FieldSpecification
|
||
open FieldOpFreeAlgebra
|
||
open HepLean.List
|
||
open FieldStatistic
|
||
|
||
variable (𝓕 : FieldSpecification)
|
||
|
||
/-- The set contains the super-commutators equal to zero in the operator algebra.
|
||
This contains e.g. the super-commutator of two creation operators. -/
|
||
def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
|
||
{ x |
|
||
(∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
|
||
x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca)
|
||
∨ (∃ (φc φc' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
|
||
x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛca)
|
||
∨ (∃ (φa φa' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
|
||
x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛca)
|
||
∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
|
||
x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
|
||
|
||
/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
|
||
of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by
|
||
- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` field creation operators.
|
||
This corresponds to the condition that two creation operators always super-commute.
|
||
- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` field annihilation operators.
|
||
This corresponds to the condition that two annihilation operators always super-commute.
|
||
- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
|
||
This corresponds to the condition that two operators with different statistics always
|
||
super-commute. In other words, fermions and bosons always super-commute.
|
||
- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
|
||
when combined with the conditions above, that the super-commutator is in the center of the
|
||
of the algebra.
|
||
-/
|
||
abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
|
||
|
||
namespace FieldOpAlgebra
|
||
variable {𝓕 : FieldSpecification}
|
||
|
||
/-- 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 : FieldOpFreeAlgebra 𝓕) :
|
||
x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||
rw [← TwoSidedIdeal.rel_iff]
|
||
rfl
|
||
|
||
lemma equiv_iff_exists_add (x y : FieldOpFreeAlgebra 𝓕) :
|
||
x ≈ y ↔ ∃ a, x = y + a ∧ a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||
apply Iff.intro
|
||
· intro h
|
||
rw [equiv_iff_sub_mem_ideal] at h
|
||
use x - y
|
||
simp [h]
|
||
· intro h
|
||
obtain ⟨a, rfl, ha⟩ := h
|
||
rw [equiv_iff_sub_mem_ideal]
|
||
simp [ha]
|
||
|
||
/-- For a field specification `𝓕`, the projection
|
||
|
||
`𝓕.FieldOpFreeAlgebra →ₐ[ℂ] FieldOpAlgebra 𝓕`
|
||
|
||
taking each element of `𝓕.FieldOpFreeAlgebra` to its equivalence class in `FieldOpAlgebra 𝓕`. -/
|
||
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
|
||
map_zero' := by rfl
|
||
map_add' x y := by rfl
|
||
commutes' x := by rfl
|
||
|
||
lemma ι_surjective : Function.Surjective (@ι 𝓕) := by
|
||
intro x
|
||
obtain ⟨x⟩ := x
|
||
use x
|
||
rfl
|
||
|
||
lemma ι_apply (x : FieldOpFreeAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
|
||
|
||
lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||
ι x = 0 := by
|
||
rw [ι_apply]
|
||
change ⟦x⟧ = ⟦0⟧
|
||
simp only [ringConGen, Quotient.eq]
|
||
refine RingConGen.Rel.of x 0 ?_
|
||
simpa using hx
|
||
|
||
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]
|
||
simp only [exists_prop]
|
||
right
|
||
left
|
||
use φc, φc', hφc, hφc'
|
||
|
||
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
|
||
simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
|
||
simp only [exists_prop]
|
||
right
|
||
right
|
||
left
|
||
use φa, φa', hφa, hφa'
|
||
|
||
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]
|
||
right
|
||
right
|
||
right
|
||
use φ, ψ
|
||
|
||
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 ⊢
|
||
rcases statistic_neq_of_superCommuteF_fermionic h with h | h
|
||
· simp only [ofCrAnListF_singleton]
|
||
apply ι_superCommuteF_of_diff_statistic
|
||
simpa using h
|
||
· simp [h]
|
||
|
||
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
|
||
· simp_all [ofCrAnListF_singleton]
|
||
· simp_all only [ofCrAnListF_singleton]
|
||
right
|
||
exact ι_superCommuteF_zero_of_fermionic _ _ h
|
||
|
||
/-!
|
||
|
||
## Super-commutes are in the center
|
||
|
||
-/
|
||
|
||
@[simp]
|
||
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.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
|
||
· rw [bonsonic_superCommuteF_symm h]
|
||
simp [ofCrAnListF_singleton]
|
||
· rcases ofCrAnListF_bosonic_or_fermionic [φ3] with h' | h'
|
||
· rw [superCommuteF_bonsonic_symm h']
|
||
simp [ofCrAnListF_singleton]
|
||
· rw [superCommuteF_fermionic_fermionic_symm h h']
|
||
simp [ofCrAnListF_singleton]
|
||
|
||
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
|
||
· rw [superCommuteF_bosonic_ofCrAnListF_eq_sum _ _ h]
|
||
simp [ofCrAnListF_singleton, ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF]
|
||
· rw [superCommuteF_fermionic_ofCrAnListF_eq_sum _ _ h]
|
||
simp [ofCrAnListF_singleton, ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF]
|
||
|
||
@[simp]
|
||
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
|
||
apply (ofCrAnListFBasis.ext fun l ↦ ?_)
|
||
simp [ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF]
|
||
rw [h1]
|
||
simp
|
||
|
||
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
|
||
swap
|
||
· simp [h]
|
||
trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca
|
||
· rw [bosonic_superCommuteF h]
|
||
simp
|
||
· simp
|
||
|
||
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
|
||
obtain ⟨a, rfl⟩ := ι_surjective a
|
||
have h0 := ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF φ ψ a
|
||
trans ι ((superCommuteF (ofCrAnOpF φ)) (ofCrAnOpF ψ)) * ι a + 0
|
||
swap
|
||
simp only [add_zero]
|
||
rw [← h0]
|
||
abel
|
||
|
||
/-!
|
||
|
||
## The kernel of ι
|
||
-/
|
||
|
||
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⟧ ↔ _
|
||
simp only [ringConGen, Quotient.eq]
|
||
rw [TwoSidedIdeal.mem_iff]
|
||
simp only
|
||
rfl
|
||
|
||
lemma bosonicProjF_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕)
|
||
(hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||
x.bosonicProjF.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProjF = 0 := by
|
||
have hx' := hx
|
||
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
|
||
rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
|
||
⟨φ, φ', hdiff, rfl⟩
|
||
· rcases superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic φ1 φ2 φ3 with h | h
|
||
· left
|
||
rw [bosonicProjF_of_mem_bosonic _ h]
|
||
simpa using hx'
|
||
· right
|
||
rw [bosonicProjF_of_mem_fermionic _ h]
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φc φc' with h | h
|
||
· left
|
||
rw [bosonicProjF_of_mem_bosonic _ h]
|
||
simpa using hx'
|
||
· right
|
||
rw [bosonicProjF_of_mem_fermionic _ h]
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φa φa' with h | h
|
||
· left
|
||
rw [bosonicProjF_of_mem_bosonic _ h]
|
||
simpa using hx'
|
||
· right
|
||
rw [bosonicProjF_of_mem_fermionic _ h]
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ φ' with h | h
|
||
· left
|
||
rw [bosonicProjF_of_mem_bosonic _ h]
|
||
simpa using hx'
|
||
· right
|
||
rw [bosonicProjF_of_mem_fermionic _ h]
|
||
|
||
lemma fermionicProjF_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕)
|
||
(hx : x ∈ 𝓕.fieldOpIdealSet) :
|
||
x.fermionicProjF.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProjF = 0 := by
|
||
have hx' := hx
|
||
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
|
||
rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
|
||
⟨φ, φ', hdiff, rfl⟩
|
||
· rcases superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic φ1 φ2 φ3 with h | h
|
||
· right
|
||
rw [fermionicProjF_of_mem_bosonic _ h]
|
||
· left
|
||
rw [fermionicProjF_of_mem_fermionic _ h]
|
||
simpa using hx'
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φc φc' with h | h
|
||
· right
|
||
rw [fermionicProjF_of_mem_bosonic _ h]
|
||
· left
|
||
rw [fermionicProjF_of_mem_fermionic _ h]
|
||
simpa using hx'
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φa φa' with h | h
|
||
· right
|
||
rw [fermionicProjF_of_mem_bosonic _ h]
|
||
· left
|
||
rw [fermionicProjF_of_mem_fermionic _ h]
|
||
simpa using hx'
|
||
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ φ' with h | h
|
||
· right
|
||
rw [fermionicProjF_of_mem_bosonic _ h]
|
||
· left
|
||
rw [fermionicProjF_of_mem_fermionic _ h]
|
||
simpa using hx'
|
||
|
||
lemma bosonicProjF_mem_ideal (x : FieldOpFreeAlgebra 𝓕)
|
||
(hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||
x.bosonicProjF.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
|
||
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕)
|
||
(h : a ∈ AddSubgroup.closure k) : Prop :=
|
||
a.bosonicProjF.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
|
||
change p x hx
|
||
apply AddSubgroup.closure_induction
|
||
· intro x hx
|
||
simp only [p]
|
||
obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
|
||
obtain ⟨d, hd, y, hy, rfl⟩ := Set.mem_mul.mp ha
|
||
rw [bosonicProjF_mul, bosonicProjF_mul, fermionicProjF_mul]
|
||
simp only [add_mul]
|
||
rcases fermionicProjF_mem_fieldOpIdealSet_or_zero y hy with hfy | hfy
|
||
<;> rcases bosonicProjF_mem_fieldOpIdealSet_or_zero y hy with hby | hby
|
||
· apply TwoSidedIdeal.add_mem
|
||
apply TwoSidedIdeal.add_mem
|
||
· /- boson, boson, boson mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProjF d) * ↑(bosonicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProjF y).1
|
||
simp [hby]
|
||
· use ↑(bosonicProjF b)
|
||
simp
|
||
· /- fermion, fermion, boson mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(fermionicProjF d) * ↑(fermionicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use fermionicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (fermionicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(bosonicProjF b)
|
||
simp
|
||
apply TwoSidedIdeal.add_mem
|
||
· /- boson, fermion, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProjF d) * ↑(fermionicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (fermionicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProjF b)
|
||
simp
|
||
· /- fermion, boson, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(fermionicProjF d) * ↑(bosonicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use fermionicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProjF b)
|
||
simp
|
||
· simp only [hby, ZeroMemClass.coe_zero, mul_zero, zero_mul, zero_add, add_zero]
|
||
apply TwoSidedIdeal.add_mem
|
||
· /- fermion, fermion, boson mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(fermionicProjF d) * ↑(fermionicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use fermionicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (fermionicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(bosonicProjF b)
|
||
simp
|
||
· /- boson, fermion, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProjF d) * ↑(fermionicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (fermionicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProjF b)
|
||
simp
|
||
· simp only [hfy, ZeroMemClass.coe_zero, mul_zero, zero_mul, add_zero, zero_add]
|
||
apply TwoSidedIdeal.add_mem
|
||
· /- boson, boson, boson mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProjF d) * ↑(bosonicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProjF y).1
|
||
simp [hby]
|
||
· use ↑(bosonicProjF b)
|
||
simp
|
||
· /- fermion, boson, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(fermionicProjF d) * ↑(bosonicProjF y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use fermionicProjF d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProjF y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProjF b)
|
||
simp
|
||
· simp [hfy, hby]
|
||
· simp [p]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all only [map_add, Submodule.coe_add, p]
|
||
apply TwoSidedIdeal.add_mem
|
||
· exact hpx
|
||
· exact hpy
|
||
· intro x hx
|
||
simp [p]
|
||
|
||
lemma fermionicProjF_mem_ideal (x : FieldOpFreeAlgebra 𝓕)
|
||
(hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||
x.fermionicProjF.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||
have hb := bosonicProjF_mem_ideal x hx
|
||
rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢
|
||
rw [← bosonicProjF_add_fermionicProjF x] at hx
|
||
simp only [map_add] at hx
|
||
simp_all
|
||
|
||
lemma ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero (x : FieldOpFreeAlgebra 𝓕) :
|
||
ι x = 0 ↔ ι x.bosonicProjF.1 = 0 ∧ ι x.fermionicProjF.1 = 0 := by
|
||
apply Iff.intro
|
||
· intro h
|
||
rw [ι_eq_zero_iff_mem_ideal] at h ⊢
|
||
rw [ι_eq_zero_iff_mem_ideal]
|
||
apply And.intro
|
||
· exact bosonicProjF_mem_ideal x h
|
||
· exact fermionicProjF_mem_ideal x h
|
||
· intro h
|
||
rw [← bosonicProjF_add_fermionicProjF x]
|
||
simp_all
|
||
|
||
/-!
|
||
|
||
## Constructors
|
||
|
||
-/
|
||
|
||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`,
|
||
`ofFieldOp φ` is defined as the element of
|
||
`𝓕.FieldOpAlgebra` given by `ι (ofFieldOpF φ)`. -/
|
||
def ofFieldOp (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
|
||
|
||
lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.FieldOp) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
|
||
|
||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.FieldOp`,
|
||
`ofFieldOpList φs` is defined as the element of
|
||
`𝓕.FieldOpAlgebra` given by `ι (ofFieldOpListF φ)`. -/
|
||
def ofFieldOpList (φs : List 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
|
||
|
||
lemma ofFieldOpList_eq_ι_ofFieldOpListF (φs : List 𝓕.FieldOp) :
|
||
ofFieldOpList φs = ι (ofFieldOpListF φs) := rfl
|
||
|
||
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.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.FieldOp) :
|
||
ofFieldOpList [φ] = ofFieldOp φ := by
|
||
simp only [ofFieldOpList, ofFieldOp, ofFieldOpListF_singleton]
|
||
|
||
/-- For a field specification `𝓕` and an element `φ` of `𝓕.CrAnFieldOp`,
|
||
`ofCrAnOp φ` is defined as the element of
|
||
`𝓕.FieldOpAlgebra` given by `ι (ofCrAnOpF φ)`. -/
|
||
def ofCrAnOp (φ : 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
|
||
|
||
lemma ofCrAnOp_eq_ι_ofCrAnOpF (φ : 𝓕.CrAnFieldOp) :
|
||
ofCrAnOp φ = ι (ofCrAnOpF φ) := rfl
|
||
|
||
lemma ofFieldOp_eq_sum (φ : 𝓕.FieldOp) :
|
||
ofFieldOp φ = (∑ i : 𝓕.fieldOpToCrAnType φ, ofCrAnOp ⟨φ, i⟩) := by
|
||
rw [ofFieldOp, ofFieldOpF]
|
||
simp only [map_sum]
|
||
rfl
|
||
|
||
/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`,
|
||
`ofCrAnList φs` is defined as the element of
|
||
`𝓕.FieldOpAlgebra` given by `ι (ofCrAnListF φ)`. -/
|
||
def ofCrAnList (φs : List 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
|
||
|
||
lemma ofCrAnList_eq_ι_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
|
||
ofCrAnList φs = ι (ofCrAnListF φs) := rfl
|
||
|
||
lemma ofCrAnList_append (φs ψs : List 𝓕.CrAnFieldOp) :
|
||
ofCrAnList (φs ++ ψs) = ofCrAnList φs * ofCrAnList ψs := by
|
||
simp only [ofCrAnList]
|
||
rw [ofCrAnListF_append]
|
||
simp
|
||
|
||
lemma ofCrAnList_singleton (φ : 𝓕.CrAnFieldOp) :
|
||
ofCrAnList [φ] = ofCrAnOp φ := by
|
||
simp only [ofCrAnList, ofCrAnOp, ofCrAnListF_singleton]
|
||
|
||
lemma ofFieldOpList_eq_sum (φs : List 𝓕.FieldOp) :
|
||
ofFieldOpList φs = ∑ s : CrAnSection φs, ofCrAnList s.1 := by
|
||
rw [ofFieldOpList, ofFieldOpListF_sum]
|
||
simp only [map_sum]
|
||
rfl
|
||
|
||
remark notation_drop := "In doc-strings we will often drop explicit applications of `ofCrAnOp`,
|
||
`ofCrAnList`, `ofFieldOp`, and `ofFieldOpList`"
|
||
|
||
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
|
||
annihilation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
|
||
Thus for `φ`
|
||
- an incoming asymptotic state this is `0`.
|
||
- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
|
||
- an outgoing asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`. -/
|
||
def anPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
|
||
|
||
lemma anPart_eq_ι_anPartF (φ : 𝓕.FieldOp) : anPart φ = ι (anPartF φ) := rfl
|
||
|
||
@[simp]
|
||
lemma anPart_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
|
||
anPart (FieldOp.inAsymp φ) = 0 := by
|
||
simp [anPart, anPartF]
|
||
|
||
@[simp]
|
||
lemma anPart_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
|
||
anPart (FieldOp.position φ) =
|
||
ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
|
||
simp [anPart, ofCrAnOp]
|
||
|
||
@[simp]
|
||
lemma anPart_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
|
||
anPart (FieldOp.outAsymp φ) = ofCrAnOp ⟨FieldOp.outAsymp φ, ()⟩ := by
|
||
simp [anPart, ofCrAnOp]
|
||
|
||
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
|
||
creation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
|
||
Thus for `φ`
|
||
- an incoming asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`.
|
||
- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
|
||
- an outgoing asymptotic state this is `0`. -/
|
||
def crPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
|
||
|
||
lemma crPart_eq_ι_crPartF (φ : 𝓕.FieldOp) : crPart φ = ι (crPartF φ) := rfl
|
||
|
||
@[simp]
|
||
lemma crPart_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
|
||
crPart (FieldOp.inAsymp φ) = ofCrAnOp ⟨FieldOp.inAsymp φ, ()⟩ := by
|
||
simp [crPart, ofCrAnOp]
|
||
|
||
@[simp]
|
||
lemma crPart_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
|
||
crPart (FieldOp.position φ) =
|
||
ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
|
||
simp [crPart, ofCrAnOp]
|
||
|
||
@[simp]
|
||
lemma crPart_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
|
||
crPart (FieldOp.outAsymp φ) = 0 := by
|
||
simp [crPart]
|
||
|
||
/-- For field specification `𝓕`, and an element `φ` of `𝓕.FieldOp` the following relation holds:
|
||
|
||
`ofFieldOp φ = crPart φ + anPart φ`
|
||
|
||
That is, every field operator splits into its creation part plus its annihilation part.
|
||
-/
|
||
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]
|
||
|
||
end FieldOpAlgebra
|
||
end FieldSpecification
|