248 lines
10 KiB
Text
248 lines
10 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.NormalOrder
|
||
import HepLean.PerturbationTheory.FieldOpAlgebra.SuperCommute
|
||
/-!
|
||
|
||
# Normal Ordering on Field operator algebra
|
||
|
||
-/
|
||
|
||
namespace FieldSpecification
|
||
open FieldOpFreeAlgebra
|
||
open HepLean.List
|
||
open FieldStatistic
|
||
|
||
namespace FieldOpAlgebra
|
||
variable {𝓕 : FieldSpecification}
|
||
|
||
/-!
|
||
|
||
## Normal order on super-commutators.
|
||
|
||
The main result of this is
|
||
`ι_normalOrderF_superCommuteF_eq_zero_mul`
|
||
which states that applying `ι` to the normal order of something containing a super-commutator
|
||
is zero.
|
||
|
||
-/
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero
|
||
(φa φa' : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
|
||
ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * ofCrAnListF φs') = 0 := by
|
||
rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa) with hφa | hφa
|
||
<;> rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa') with hφa' | hφa'
|
||
· rw [normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF φa φa' hφa hφa' φs φs']
|
||
rw [map_smul, map_mul, map_mul, map_mul, ι_superCommuteF_of_create_create φa φa' hφa hφa']
|
||
simp
|
||
· rw [normalOrderF_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnListF φs)
|
||
(ofCrAnListF φs')]
|
||
simp
|
||
· rw [normalOrderF_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnListF φs)
|
||
(ofCrAnListF φs')]
|
||
simp
|
||
· rw [normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
|
||
φa φa' hφa hφa' φs φs']
|
||
rw [map_smul, map_mul, map_mul, map_mul,
|
||
ι_superCommuteF_of_annihilate_annihilate φa φa' hφa hφa']
|
||
simp
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero
|
||
(φa φa' : 𝓕.CrAnFieldOp) (φs : List 𝓕.CrAnFieldOp)
|
||
(a : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * a) = 0 := by
|
||
have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap (ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca) = 0 := by
|
||
apply ofCrAnListFBasis.ext
|
||
intro l
|
||
simp only [FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
|
||
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
|
||
exact ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero φa φa' φs l
|
||
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap ((ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca))) a = 0
|
||
rw [hf]
|
||
simp
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul (φa φa' : 𝓕.CrAnFieldOp)
|
||
(a b : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b) = 0 := by
|
||
rw [mul_assoc]
|
||
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
|
||
([ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b)) a = 0
|
||
have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
|
||
([ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b) = 0 := by
|
||
apply ofCrAnListFBasis.ext
|
||
intro l
|
||
simp only [mulLinearMap, FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
|
||
Function.comp_apply, LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk,
|
||
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
|
||
rw [← mul_assoc]
|
||
exact ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero φa φa' _ _
|
||
rw [hf]
|
||
simp
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
|
||
(φs : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnListF φs]ₛca * b) = 0 := by
|
||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
|
||
rw [Finset.mul_sum, Finset.sum_mul]
|
||
rw [map_sum, map_sum]
|
||
apply Fintype.sum_eq_zero
|
||
intro n
|
||
rw [← mul_assoc, ← mul_assoc]
|
||
rw [mul_assoc _ _ b, ofCrAnListF_singleton]
|
||
rw [ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul]
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
|
||
(φs : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnOpF φa]ₛca * b) = 0 := by
|
||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_symm, ofCrAnListF_singleton]
|
||
simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
|
||
Algebra.mul_smul_comm, neg_mul, Algebra.smul_mul_assoc, map_neg, map_smul]
|
||
rw [ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul]
|
||
simp
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul
|
||
(φs φs' : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnListF φs']ₛca * b) = 0 := by
|
||
rw [superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum, Finset.mul_sum, Finset.sum_mul]
|
||
rw [map_sum, map_sum]
|
||
apply Fintype.sum_eq_zero
|
||
intro n
|
||
rw [← mul_assoc, ← mul_assoc]
|
||
rw [mul_assoc _ _ b]
|
||
rw [ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul]
|
||
|
||
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
|
||
(φs : List 𝓕.CrAnFieldOp)
|
||
(a b c : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [ofCrAnListF φs, c]ₛca * b) = 0 := by
|
||
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnListF φs)) c = 0
|
||
have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnListF φs)) = 0 := by
|
||
apply ofCrAnListFBasis.ext
|
||
intro φs'
|
||
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
|
||
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
|
||
LinearMap.zero_apply]
|
||
rw [ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul]
|
||
rw [hf]
|
||
simp
|
||
|
||
@[simp]
|
||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul
|
||
(a b c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
|
||
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
|
||
have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
|
||
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
|
||
apply ofCrAnListFBasis.ext
|
||
intro φs
|
||
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
|
||
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
|
||
LinearMap.zero_apply]
|
||
rw [ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul]
|
||
rw [hf]
|
||
simp
|
||
|
||
@[simp]
|
||
lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
|
||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
|
||
simp
|
||
|
||
@[simp]
|
||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
|
||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
|
||
simp
|
||
|
||
@[simp]
|
||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.FieldOpFreeAlgebra) :
|
||
ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
|
||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d]
|
||
congr 2
|
||
noncomm_ring
|
||
|
||
@[simp]
|
||
lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
|
||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
|
||
simp
|
||
|
||
/-!
|
||
|
||
## Defining normal order for `FiedOpAlgebra`.
|
||
|
||
-/
|
||
|
||
lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
|
||
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝ᶠ(a) = 0 := by
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
|
||
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕)
|
||
(h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
|
||
change p a h
|
||
apply AddSubgroup.closure_induction
|
||
· intro x hx
|
||
obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
|
||
obtain ⟨a, ha, c, hc, rfl⟩ := ha
|
||
simp only [p]
|
||
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq] at hc
|
||
match hc with
|
||
| Or.inl hc =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
simp [mul_sub, sub_mul, ← mul_assoc]
|
||
| Or.inr (Or.inl hc) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
simp [mul_sub, sub_mul, ← mul_assoc]
|
||
| Or.inr (Or.inr (Or.inl hc)) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
simp [mul_sub, sub_mul, ← mul_assoc]
|
||
| Or.inr (Or.inr (Or.inr hc)) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
simp [mul_sub, sub_mul, ← mul_assoc]
|
||
· simp [p]
|
||
· intro x y hx hy
|
||
simp only [map_add, p]
|
||
intro h1 h2
|
||
simp [h1, h2]
|
||
· intro x hx
|
||
simp [p]
|
||
|
||
lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
|
||
ι 𝓝ᶠ(a) = ι 𝓝ᶠ(b) := by
|
||
rw [equiv_iff_sub_mem_ideal] at h
|
||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||
simp only [LinearMap.mem_ker, ← map_sub]
|
||
exact ι_normalOrderF_zero_of_mem_ideal (a - b) h
|
||
|
||
/-- For a field specification `𝓕`, `normalOrder` is the linear map
|
||
|
||
`FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
|
||
|
||
defined as the decent of `ι ∘ₗ normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
|
||
from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
|
||
This decent exists because `ι ∘ₗ normalOrderF` is well-defined on equivalence classes.
|
||
|
||
The notation `𝓝(a)` is used for `normalOrder a` for `a` an element of `FieldOpAlgebra 𝓕`. -/
|
||
noncomputable def normalOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ normalOrderF) ι_normalOrderF_eq_of_equiv
|
||
map_add' x y := by
|
||
obtain ⟨x, rfl⟩ := ι_surjective x
|
||
obtain ⟨y, rfl⟩ := ι_surjective y
|
||
rw [← map_add, ι_apply, ι_apply, ι_apply]
|
||
rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
|
||
simp
|
||
map_smul' c y := by
|
||
obtain ⟨y, rfl⟩ := ι_surjective y
|
||
rw [← map_smul, ι_apply, ι_apply]
|
||
simp
|
||
|
||
@[inherit_doc normalOrder]
|
||
scoped[FieldSpecification.FieldOpAlgebra] notation "𝓝(" a ")" => normalOrder a
|
||
|
||
end FieldOpAlgebra
|
||
end FieldSpecification
|