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refactor: Rename CrAnAlgebra

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jstoobysmith 2025-02-03 11:05:43 +00:00
parent 9a5676e134
commit b0735a1e13
16 changed files with 214 additions and 214 deletions
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44
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/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.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

42
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.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.Algebras.CrAnAlgebra.NormalOrder
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.NormalOrder
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
/-!
@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
-/
namespace FieldSpecification
open CrAnAlgebra
open FieldOpFreeAlgebra
open HepLean.List
open FieldStatistic
@ -52,12 +52,12 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
(a : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
mulLinearMap (ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca) = 0 := by
apply ofCrAnListBasis.ext
intro l
simp only [CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
simp only [FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
exact ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
@ -66,7 +66,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
simp
lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
(a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
rw [mul_assoc]
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
@ -75,7 +75,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
([ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
apply ofCrAnListBasis.ext
intro l
simp only [mulLinearMap, CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
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]
@ -85,7 +85,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
(a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
rw [Finset.mul_sum, Finset.sum_mul]
@ -97,7 +97,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa :
rw [ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul]
lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
(φs : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
@ -106,7 +106,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa :
simp
lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
@ -118,7 +118,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
(φs : List 𝓕.CrAnStates)
(a b c : 𝓕.CrAnAlgebra) :
(a b c : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
@ -126,7 +126,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
apply ofCrAnListBasis.ext
intro φs'
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
@ -135,14 +135,14 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
@[simp]
lemma ι_normalOrderF_superCommuteF_eq_zero_mul
(a b c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
(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 ofCrAnListBasis.ext
intro φs
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul]
@ -150,26 +150,26 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
simp
@[simp]
lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
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.CrAnAlgebra) :
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.CrAnAlgebra) :
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.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
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
@ -179,10 +179,10 @@ lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝
-/
lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
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
@ -211,7 +211,7 @@ lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
· intro x hx
simp [p]
lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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]
@ -241,7 +241,7 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓝(" a ")" => normalOrder
-/
lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.CrAnAlgebra) :
lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.FieldOpFreeAlgebra) :
𝓝(ι a) = ι 𝓝ᶠ(a) := rfl
lemma normalOrder_ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) :

2
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/StaticWickTheorem.lean
View file

@ -14,7 +14,7 @@ import HepLean.Meta.Remark.Basic
namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open CrAnAlgebra
open FieldOpFreeAlgebra
open HepLean.List
open WickContraction

22
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/SuperCommute.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.Algebras.CrAnAlgebra.TimeOrder
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.TimeOrder
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
/-!
@ -12,20 +12,20 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
-/
namespace FieldSpecification
open CrAnAlgebra
open FieldOpFreeAlgebra
open HepLean.List
open FieldStatistic
namespace FieldOpAlgebra
variable {𝓕 : FieldSpecification}
lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι b = 0) :
ι [a, b]ₛca = 0 := by
rw [superCommuteF_expand_bosonicProj_fermionicProj]
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
simp_all
lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι a = 0) :
ι [a, b]ₛca = 0 := by
rw [superCommuteF_expand_bosonicProj_fermionicProj]
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
@ -37,12 +37,12 @@ lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι
-/
lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.FieldOpFreeAlgebra)
(h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
apply ι_superCommuteF_eq_zero_of_ι_right_zero
exact (ι_eq_zero_iff_mem_ideal b).mpr h
lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h : b1 ≈ b2) :
ι [a, b1]ₛca = ι [a, b2]ₛca := by
rw [equiv_iff_sub_mem_ideal] at h
rw [LinearMap.sub_mem_ker_iff.mp]
@ -50,7 +50,7 @@ lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1
exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
(ι_superCommuteF_eq_of_equiv_right a)
@ -67,13 +67,13 @@ noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
rw [← map_smul, ι_apply, ι_apply]
simp
lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) :
lemma superCommuteRight_apply_ι (a b : 𝓕.FieldOpFreeAlgebra) :
superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) :
lemma superCommuteRight_apply_quot (a b : 𝓕.FieldOpFreeAlgebra) :
superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.FieldOpFreeAlgebra) (h : a1 ≈ a2) :
superCommuteRight a1 = superCommuteRight a2 := by
rw [equiv_iff_sub_mem_ideal] at h
ext b
@ -114,7 +114,7 @@ noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
@[inherit_doc superCommute]
scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCommute a b
lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.CrAnAlgebra) :
lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.FieldOpFreeAlgebra) :
[ι a, ι b]ₛ = ι [a, b]ₛca := rfl
/-!

2
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeContraction.lean
View file

@ -16,7 +16,7 @@ generated by the states.
namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open CrAnAlgebra
open FieldOpFreeAlgebra
noncomputable section
namespace FieldOpAlgebra

26
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.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.Algebras.CrAnAlgebra.TimeOrder
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.TimeOrder
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
/-!
@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
-/
namespace FieldSpecification
open CrAnAlgebra
open FieldOpFreeAlgebra
open HepLean.List
open FieldStatistic
@ -153,16 +153,16 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.C
simp
@[simp]
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
@ -182,10 +182,10 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates}
simp_all [pb, hpx]
lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
change pb b (Basis.mem_span _ b)
@ -193,7 +193,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, map_mul, pb]
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
change pa a (Basis.mem_span _ a)
@ -278,7 +278,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
simp_all [pb, hpx]
lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
rw [timeOrderF_timeOrderF_mid]
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
@ -300,10 +300,10 @@ lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
-/
lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
change p a h
apply AddSubgroup.closure_induction
· intro x hx
@ -358,7 +358,7 @@ lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
· intro x hx
simp [p]
lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
ι 𝓣ᶠ(a) = ι 𝓣ᶠ(b) := by
rw [equiv_iff_sub_mem_ideal] at h
rw [LinearMap.sub_mem_ker_iff.mp]
@ -390,7 +390,7 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓣(" a ")" => timeOrder a
-/
lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.CrAnAlgebra) :
lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.FieldOpFreeAlgebra) :
𝓣(ι a) = ι 𝓣ᶠ(a) := rfl
lemma timeOrder_ofFieldOp_ofFieldOp_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :

2
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/WicksTheoremNormal.lean
View file

@ -15,7 +15,7 @@ import HepLean.Meta.Remark.Basic
namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open CrAnAlgebra
open FieldOpFreeAlgebra
namespace FieldOpAlgebra
open WickContraction
open EqTimeOnly

38
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Basic.lean → HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Basic.lean
View file

@ -18,7 +18,7 @@ The algebra is spanned by lists of creation/annihilation states.
The main structures defined in this module are:
* `CrAnAlgebra` - The creation and annihilation algebra
* `FieldOpFreeAlgebra` - The creation and annihilation algebra
* `ofCrAnState` - Maps a creation/annihilation state to the algebra
* `ofCrAnList` - Maps a list of creation/annihilation states to the algebra
* `ofState` - Maps a state to a sum of creation and annihilation operators
@ -38,18 +38,18 @@ variable {𝓕 : FieldSpecification}
The free algebra generated by `CrAnStates`,
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 `CrAnAlgebra` is spanned by lists of `CrAnStates`. -/
abbrev CrAnAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnStates
As a module `FieldOpFreeAlgebra` is spanned by lists of `CrAnStates`. -/
abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnStates
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
/-- Maps a creation and annihlation state to the creation and annihlation free-algebra. -/
def ofCrAnState (φ : 𝓕.CrAnStates) : CrAnAlgebra 𝓕 :=
def ofCrAnState (φ : 𝓕.CrAnStates) : 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 ofCrAnList (φs : List 𝓕.CrAnStates) : CrAnAlgebra 𝓕 := (List.map ofCrAnState φs).prod
def ofCrAnList (φs : List 𝓕.CrAnStates) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnState φs).prod
@[simp]
lemma ofCrAnList_nil : ofCrAnList ([] : List 𝓕.CrAnStates) = 1 := rfl
@ -66,16 +66,16 @@ lemma ofCrAnList_singleton (φ : 𝓕.CrAnStates) :
/-- Maps a state to the sum of creation and annihilation operators in
creation and annihilation free-algebra. -/
def ofState (φ : 𝓕.States) : CrAnAlgebra 𝓕 :=
def ofState (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.statesToCrAnType φ), ofCrAnState ⟨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 ofStateList (φs : List 𝓕.States) : CrAnAlgebra 𝓕 := (List.map ofState φs).prod
def ofStateList (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofState φs).prod
/-- Coercion from `List 𝓕.States` to `CrAnAlgebra 𝓕` through `ofStateList`. -/
instance : Coe (List 𝓕.States) (CrAnAlgebra 𝓕) := ⟨ofStateList⟩
/-- Coercion from `List 𝓕.States` to `FieldOpFreeAlgebra 𝓕` through `ofStateList`. -/
instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofStateList⟩
@[simp]
lemma ofStateList_nil : ofStateList ([] : List 𝓕.States) = 1 := rfl
@ -113,7 +113,7 @@ lemma ofStateList_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.CrAnAlgebra := fun φ =>
def crPartF : 𝓕.States → 𝓕.FieldOpFreeAlgebra := fun φ =>
match φ with
| States.inAsymp φ => ofCrAnState ⟨States.inAsymp φ, ()⟩
| States.position φ => ofCrAnState ⟨States.position φ, CreateAnnihilate.create⟩
@ -138,7 +138,7 @@ lemma crPartF_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
/-- 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.CrAnAlgebra := fun φ =>
def anPartF : 𝓕.States → 𝓕.FieldOpFreeAlgebra := fun φ =>
match φ with
| States.inAsymp _ => 0
| States.position φ => ofCrAnState ⟨States.position φ, CreateAnnihilate.annihilate⟩
@ -175,7 +175,7 @@ lemma ofState_eq_crPartF_add_anPartF (φ : 𝓕.States) :
-/
/-- The basis of the free creation and annihilation algebra formed by lists of CrAnStates. -/
noncomputable def ofCrAnListBasis : Basis (List 𝓕.CrAnStates) ℂ (CrAnAlgebra 𝓕) where
noncomputable def ofCrAnListBasis : Basis (List 𝓕.CrAnStates) ℂ (FieldOpFreeAlgebra 𝓕) where
repr := FreeAlgebra.equivMonoidAlgebraFreeMonoid.toLinearEquiv
@[simp]
@ -197,8 +197,8 @@ lemma ofListBasis_eq_ofList (φs : List 𝓕.CrAnStates) :
-/
/-- The bi-linear map associated with multiplication in `CrAnAlgebra`. -/
noncomputable def mulLinearMap : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 where
/-- The bi-linear map associated with multiplication in `FieldOpFreeAlgebra`. -/
noncomputable def mulLinearMap : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 where
toFun a := {
toFun := fun b => a * b,
map_add' := fun c d => by simp [mul_add]
@ -211,15 +211,15 @@ noncomputable def mulLinearMap : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕
ext c
simp [smul_mul']
lemma mulLinearMap_apply (a b : CrAnAlgebra 𝓕) :
lemma mulLinearMap_apply (a b : FieldOpFreeAlgebra 𝓕) :
mulLinearMap a b = a * b := rfl
/-- The linear map associated with scalar-multiplication in `CrAnAlgebra`. -/
noncomputable def smulLinearMap (c : ℂ) : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 where
/-- The linear map associated with scalar-multiplication in `FieldOpFreeAlgebra`. -/
noncomputable def smulLinearMap (c : ℂ) : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 where
toFun a := c • a
map_add' := by simp
map_smul' m x := by simp [smul_smul, RingHom.id_apply, NonUnitalNormedCommRing.mul_comm]
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

72
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Grading.lean → HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Grading.lean
View file

@ -3,12 +3,12 @@ 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.Basic
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.Basic
import HepLean.PerturbationTheory.Koszul.KoszulSign
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Grading on the CrAnAlgebra
# Grading on the FieldOpFreeAlgebra
-/
@ -16,12 +16,12 @@ namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open FieldStatistic
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
noncomputable section
/-- The submodule of `CrAnAlgebra` spanned by lists of field statistic `f`. -/
def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
/-- The submodule of `FieldOpFreeAlgebra` spanned by lists of field statistic `f`. -/
def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.FieldOpFreeAlgebra :=
Submodule.span ℂ {a | ∃ φs, a = ofCrAnList φs ∧ (𝓕 |>ₛ φs) = f}
lemma ofCrAnList_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
@ -42,8 +42,8 @@ lemma ofCrAnState_bosonic_or_fermionic (φ : 𝓕.CrAnStates) :
rw [← ofCrAnList_singleton]
exact ofCrAnList_bosonic_or_fermionic [φ]
/-- The projection of an element of `CrAnAlgebra` onto it's bosonic part. -/
def bosonicProj : 𝓕.CrAnAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) bosonic :=
/-- The projection of an element of `FieldOpFreeAlgebra` onto it's bosonic part. -/
def bosonicProj : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) bosonic :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
if h : (𝓕 |>ₛ φs) = bosonic then
⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩
@ -56,9 +56,9 @@ lemma bosonicProj_ofCrAnList (φs : List 𝓕.CrAnStates) :
conv_lhs =>
rw [← ofListBasis_eq_ofList, bosonicProj, Basis.constr_basis]
lemma bosonicProj_of_mem_bosonic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule bosonic) :
lemma bosonicProj_of_mem_bosonic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule bosonic) :
bosonicProj a = ⟨a, h⟩ := by
let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
bosonicProj a = ⟨a, hx⟩
change p a h
apply Submodule.span_induction
@ -73,9 +73,9 @@ lemma bosonicProj_of_mem_bosonic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubm
· intro a x hx hy
simp_all [p]
lemma bosonicProj_of_mem_fermionic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule fermionic) :
lemma bosonicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule fermionic) :
bosonicProj a = 0 := by
let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
bosonicProj a = 0
change p a h
apply Submodule.span_induction
@ -102,8 +102,8 @@ lemma bosonicProj_of_fermionic_part
apply bosonicProj_of_mem_fermionic
exact Submodule.coe_mem (a.toFun fermionic)
/-- The projection of an element of `CrAnAlgebra` onto it's fermionic part. -/
def fermionicProj : 𝓕.CrAnAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) fermionic :=
/-- The projection of an element of `FieldOpFreeAlgebra` onto it's fermionic part. -/
def fermionicProj : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) fermionic :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
if h : (𝓕 |>ₛ φs) = fermionic then
⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩
@ -127,9 +127,9 @@ lemma fermionicProj_ofCrAnList_if_bosonic (φs : List 𝓕.CrAnStates) :
simp only [neq_fermionic_iff_eq_bosonic] at h1
simp [h1]
lemma fermionicProj_of_mem_fermionic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule fermionic) :
lemma fermionicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule fermionic) :
fermionicProj a = ⟨a, h⟩ := by
let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
fermionicProj a = ⟨a, hx⟩
change p a h
apply Submodule.span_induction
@ -144,9 +144,9 @@ lemma fermionicProj_of_mem_fermionic (a : 𝓕.CrAnAlgebra) (h : a ∈ statistic
· intro a x hx hy
simp_all [p]
lemma fermionicProj_of_mem_bosonic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule bosonic) :
lemma fermionicProj_of_mem_bosonic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule bosonic) :
fermionicProj a = 0 := by
let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
fermionicProj a = 0
change p a h
apply Submodule.span_induction
@ -173,11 +173,11 @@ lemma fermionicProj_of_fermionic_part
fermionicProj (a fermionic) = a fermionic := by
apply fermionicProj_of_mem_fermionic
lemma bosonicProj_add_fermionicProj (a : 𝓕.CrAnAlgebra) :
lemma bosonicProj_add_fermionicProj (a : 𝓕.FieldOpFreeAlgebra) :
a.bosonicProj + (a.fermionicProj).1 = a := by
let f1 :𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
let f1 :𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra :=
(statisticSubmodule bosonic).subtype ∘ₗ bosonicProj
let f2 :𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
let f2 :𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra :=
(statisticSubmodule fermionic).subtype ∘ₗ fermionicProj
change (f1 + f2) a = LinearMap.id (R := ℂ) a
refine LinearMap.congr_fun (ofCrAnListBasis.ext fun φs ↦ ?_) a
@ -234,8 +234,8 @@ lemma directSum_eq_bosonic_plus_fermionic
conv_lhs => rw [hx, hy]
abel
/-- The instance of a graded algebra on `CrAnAlgebra`. -/
instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
/-- The instance of a graded algebra on `FieldOpFreeAlgebra`. -/
instance fieldOpFreeAlgebraGrade : GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where
one_mem := by
simp only [statisticSubmodule]
refine Submodule.mem_span.mpr fun p a => a ?_
@ -244,7 +244,7 @@ instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmo
simp only [ofCrAnList_nil, ofList_empty, true_and]
rfl
mul_mem f1 f2 a1 a2 h1 h2 := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
a1 * a2 ∈ statisticSubmodule (f1 + f2)
change p a2 h2
apply Submodule.span_induction (p := p)
@ -252,7 +252,7 @@ instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmo
simp only [Set.mem_setOf_eq] at hx
obtain ⟨φs, rfl, h⟩ := hx
simp only [p]
let p (a1 : 𝓕.CrAnAlgebra) (hx : a1 ∈ statisticSubmodule f1) : Prop :=
let p (a1 : 𝓕.FieldOpFreeAlgebra) (hx : a1 ∈ statisticSubmodule f1) : Prop :=
a1 * ofCrAnList φs ∈ statisticSubmodule (f1 + f2)
change p a1 h1
apply Submodule.span_induction (p := p)
@ -294,7 +294,7 @@ instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmo
fermionicProj_of_fermionic_part, zero_add]
conv_rhs => rw [directSum_eq_bosonic_plus_fermionic a]
lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.CrAnAlgebra}
lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.FieldOpFreeAlgebra}
(hb : a ∈ statisticSubmodule bosonic) (hf : a ∈ statisticSubmodule fermionic) : a = 0 := by
have ha := bosonicProj_of_mem_bosonic a hb
have hb := fermionicProj_of_mem_fermionic a hf
@ -302,7 +302,7 @@ lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.CrAnAlgebra}
rw [ha, hb] at hc
simpa using hc
lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
lemma bosonicProj_mul (a b : 𝓕.FieldOpFreeAlgebra) :
(a * b).bosonicProj.1 = a.bosonicProj.1 * b.bosonicProj.1
+ a.fermionicProj.1 * b.fermionicProj.1 := by
conv_lhs =>
@ -317,7 +317,7 @@ lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
(by
have h1 : fermionic = fermionic + bosonic := by simp
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
conv_lhs =>
@ -327,7 +327,7 @@ lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
(by
have h1 : fermionic = bosonic + fermionic := by simp
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
conv_lhs =>
@ -339,7 +339,7 @@ lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
simp only [ZeroMemClass.coe_zero, add_zero, zero_add]
@ -347,11 +347,11 @@ lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp
lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
lemma fermionicProj_mul (a b : 𝓕.FieldOpFreeAlgebra) :
(a * b).fermionicProj.1 = a.bosonicProj.1 * b.fermionicProj.1
+ a.fermionicProj.1 * b.bosonicProj.1 := by
conv_lhs =>
@ -367,7 +367,7 @@ lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
conv_lhs =>
@ -377,7 +377,7 @@ lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
(by
have h1 : fermionic = fermionic + bosonic := by simp
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
conv_lhs =>
@ -387,7 +387,7 @@ lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
(by
have h1 : fermionic = bosonic + fermionic := by simp
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
conv_lhs =>
@ -399,7 +399,7 @@ lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
conv_lhs => rw [h1]
apply crAnAlgebraGrade.mul_mem
apply fieldOpFreeAlgebraGrade.mul_mem
simp only [SetLike.coe_mem]
simp)]
simp only [ZeroMemClass.coe_zero, zero_add, add_zero]
@ -407,6 +407,6 @@ lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
end
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

14
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/NormTimeOrder.lean → HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormTimeOrder.lean
View file

@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.PerturbationTheory.FieldSpecification.TimeOrder
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.SuperCommute
import HepLean.PerturbationTheory.Koszul.KoszulSign
/-!
# Norm-time Ordering in the CrAnAlgebra
# Norm-time Ordering in the FieldOpFreeAlgebra
-/
@ -16,7 +16,7 @@ namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open FieldStatistic
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
noncomputable section
open HepLean.List
@ -27,13 +27,13 @@ open HepLean.List
-/
/-- The normal-time ordering on `CrAnAlgebra`. -/
def normTimeOrder : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
/-- The normal-time ordering on `FieldOpFreeAlgebra`. -/
def normTimeOrder : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
normTimeOrderSign φs • ofCrAnList (normTimeOrderList φs)
@[inherit_doc normTimeOrder]
scoped[FieldSpecification.CrAnAlgebra] notation "𝓣𝓝ᶠ(" a ")" => normTimeOrder a
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓣𝓝ᶠ(" a ")" => normTimeOrder a
lemma normTimeOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
𝓣𝓝ᶠ(ofCrAnList φs) = normTimeOrderSign φs • ofCrAnList (normTimeOrderList φs) := by
@ -42,6 +42,6 @@ lemma normTimeOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
end
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

50
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/NormalOrder.lean → HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormalOrder.lean
View file

@ -4,16 +4,16 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.PerturbationTheory.FieldSpecification.NormalOrder
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.SuperCommute
import HepLean.PerturbationTheory.Koszul.KoszulSign
/-!
# Normal Ordering in the CrAnAlgebra
# Normal Ordering in the FieldOpFreeAlgebra
In the module
`HepLean.PerturbationTheory.FieldSpecification.NormalOrder`
we defined the normal ordering of a list of `CrAnStates`.
In this module we extend the normal ordering to a linear map on `CrAnAlgebra`.
In this module we extend the normal ordering to a linear map on `FieldOpFreeAlgebra`.
We derive properties of this normal ordering.
@ -23,7 +23,7 @@ namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open FieldStatistic
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
noncomputable section
@ -32,12 +32,12 @@ noncomputable section
a list of CrAnStates to the normal-ordered list of states multiplied by
the sign corresponding to the number of fermionic-fermionic
exchanges done in ordering. -/
def normalOrderF : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
normalOrderSign φs • ofCrAnList (normalOrderList φs)
@[inherit_doc normalOrderF]
scoped[FieldSpecification.CrAnAlgebra] notation "𝓝ᶠ(" a ")" => normalOrderF a
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓝ᶠ(" a ")" => normalOrderF a
lemma normalOrderF_ofCrAnList (φs : List 𝓕.CrAnStates) :
𝓝ᶠ(ofCrAnList φs) = normalOrderSign φs • ofCrAnList (normalOrderList φs) := by
@ -52,23 +52,23 @@ lemma normalOrderF_one : normalOrderF (𝓕 := 𝓕) 1 = 1 := by
rw [← ofCrAnList_nil, normalOrderF_ofCrAnList, normalOrderSign_nil, normalOrderList_nil,
ofCrAnList_nil, one_smul]
lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.CrAnAlgebra) :
lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * b * c) = 𝓝ᶠ(a * 𝓝ᶠ(b) * c) := by
let pc (c : 𝓕.CrAnAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pc (c : 𝓕.FieldOpFreeAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓝ᶠ(a * b * c) = 𝓝ᶠ(a * 𝓝ᶠ(b) * c)
change pc c (Basis.mem_span _ c)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pc]
let pb (b : 𝓕.CrAnAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pb (b : 𝓕.FieldOpFreeAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓝ᶠ(a * b * ofCrAnList φs) = 𝓝ᶠ(a * 𝓝ᶠ(b) * ofCrAnList φs)
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.CrAnAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pa (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓝ᶠ(a * ofCrAnList φs' * ofCrAnList φs) = 𝓝ᶠ(a * 𝓝ᶠ(ofCrAnList φs') * ofCrAnList φs)
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
@ -101,13 +101,13 @@ lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.CrAnAlgebra) :
· intro x hx h hp
simp_all [pc]
lemma normalOrderF_normalOrderF_right (a b : 𝓕.CrAnAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by
lemma normalOrderF_normalOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by
trans 𝓝ᶠ(a * b * 1)
· simp
· rw [normalOrderF_normalOrderF_mid]
simp
lemma normalOrderF_normalOrderF_left (a b : 𝓕.CrAnAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by
lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by
trans 𝓝ᶠ(1 * a * b)
· simp
· rw [normalOrderF_normalOrderF_mid]
@ -127,7 +127,7 @@ lemma normalOrderF_ofCrAnList_cons_create (φ : 𝓕.CrAnStates)
rw [ofCrAnList_cons, normalOrderF_ofCrAnList, mul_smul_comm]
lemma normalOrderF_create_mul (φ : 𝓕.CrAnStates)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (a : CrAnAlgebra 𝓕) :
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(ofCrAnState φ * a) = ofCrAnState φ * 𝓝ᶠ(a) := by
change (normalOrderF ∘ₗ mulLinearMap (ofCrAnState φ)) a =
(mulLinearMap (ofCrAnState φ) ∘ₗ normalOrderF) a
@ -145,7 +145,7 @@ lemma normalOrderF_ofCrAnList_append_annihilate (φ : 𝓕.CrAnStates)
lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnStates)
(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate)
(a : CrAnAlgebra 𝓕) : 𝓝ᶠ(a * ofCrAnState φ) = 𝓝ᶠ(a) * ofCrAnState φ := by
(a : FieldOpFreeAlgebra 𝓕) : 𝓝ᶠ(a * ofCrAnState φ) = 𝓝ᶠ(a) * ofCrAnState φ := by
change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnState φ)) a =
(mulLinearMap.flip (ofCrAnState φ) ∘ₗ normalOrderF) a
refine LinearMap.congr_fun (ofCrAnListBasis.ext fun l ↦ ?_) a
@ -154,7 +154,7 @@ lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnStates)
rw [← ofCrAnList_singleton, ← ofCrAnList_append, ofCrAnList_singleton,
normalOrderF_ofCrAnList_append_annihilate φ hφ]
lemma normalOrderF_crPartF_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
lemma normalOrderF_crPartF_mul (φ : 𝓕.States) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(crPartF φ * a) =
crPartF φ * 𝓝ᶠ(a) := by
match φ with
@ -166,7 +166,7 @@ lemma normalOrderF_crPartF_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
exact normalOrderF_create_mul _ rfl _
| .outAsymp φ => simp
lemma normalOrderF_mul_anPartF (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
lemma normalOrderF_mul_anPartF (φ : 𝓕.States) (a : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * anPartF φ) =
𝓝ᶠ(a) * anPartF φ := by
match φ with
@ -198,7 +198,7 @@ lemma normalOrderF_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.C
lemma normalOrderF_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(φs : List 𝓕.CrAnStates) (a : 𝓕.CrAnAlgebra) :
(φs : List 𝓕.CrAnStates) (a : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(ofCrAnList φs * ofCrAnState φc * ofCrAnState φa * a) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
𝓝ᶠ(ofCrAnList φs * ofCrAnState φa * ofCrAnState φc * a) := by
change (normalOrderF ∘ₗ mulLinearMap (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa)) a =
@ -212,7 +212,7 @@ lemma normalOrderF_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.CrAnAlgebra) :
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * ofCrAnState φc * ofCrAnState φa * b) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
𝓝ᶠ(a * ofCrAnState φa * ofCrAnState φc * b) := by
rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc]
@ -227,7 +227,7 @@ lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.CrAnAlgebra) :
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnState φc, ofCrAnState φa]ₛca * b) = 0 := by
simp only [superCommuteF_ofCrAnState_ofCrAnState, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc, ← mul_assoc, ← mul_assoc,
@ -236,14 +236,14 @@ lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.CrAnAlgebra) :
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φc]ₛca * b) = 0 := by
rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
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 : CrAnAlgebra 𝓕) :
lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.States) (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
@ -279,13 +279,13 @@ Using the results from above.
-/
lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.States) (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 : CrAnAlgebra 𝓕) :
lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) (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 : C
rw [crPartF_position, anPartF_posAsymp]
exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
lemma normalOrderF_superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) (a b : FieldOpFreeAlgebra 𝓕) :
𝓝ᶠ(a * superCommuteF
(anPartF φ) (crPartF φ') * b) = 0 := by
match φ, φ' with
@ -570,6 +570,6 @@ lemma anPartF_mul_normalOrderF_ofStateList_eq_superCommuteF (φ : 𝓕.States)
end
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

64
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean → HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/SuperCommute.lean
View file

@ -3,8 +3,8 @@ 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.Basic
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Grading
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.Basic
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.Grading
/-!
# Super Commute
@ -13,11 +13,11 @@ import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Grading
namespace FieldSpecification
variable {𝓕 : FieldSpecification}
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
/-!
## The super commutor on the CrAnAlgebra.
## The super commutor on the FieldOpFreeAlgebra.
-/
@ -26,7 +26,7 @@ open FieldStatistic
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
noncomputable def superCommuteF : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
Basis.constr ofCrAnListBasis ℂ fun φs' =>
ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs)
@ -34,7 +34,7 @@ noncomputable def superCommuteF : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
scoped[FieldSpecification.CrAnAlgebra] notation "[" φs "," φs' "]ₛca" => superCommuteF φs φs'
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca" => superCommuteF φs φs'
/-!
@ -486,17 +486,17 @@ lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
-/
lemma superCommuteF_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatistic}
(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
[a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
simp only [add_eq_mul, instCommGroup, p]
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
[a2, ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
change p a ha
apply Submodule.span_induction (p := p)
@ -528,16 +528,16 @@ lemma superCommuteF_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
exact Submodule.smul_mem _ c hp1
· exact hb
lemma superCommuteF_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
@ -561,16 +561,16 @@ lemma superCommuteF_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommuteF_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
@ -594,16 +594,16 @@ lemma superCommuteF_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
@ -627,7 +627,7 @@ lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommuteF_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
@ -635,7 +635,7 @@ lemma superCommuteF_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmo
simp only [add_mul, mul_add]
abel
lemma bosonic_superCommuteF {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj b]
simp only [map_add, LinearMap.add_apply]
@ -643,26 +643,26 @@ lemma bosonic_superCommuteF {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmod
simp only [add_mul, mul_add]
abel
lemma superCommuteF_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
simp
lemma bonsonic_superCommuteF_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
simp
lemma superCommuteF_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b + b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a, a2]ₛca = a * a2 + a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
@ -686,14 +686,14 @@ lemma superCommuteF_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = [b, a]ₛca := by
rw [superCommuteF_fermionic_fermionic ha hb]
rw [superCommuteF_fermionic_fermionic hb ha]
abel
lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.FieldOpFreeAlgebra) :
[a, b]ₛca = bosonicProj a * bosonicProj b - bosonicProj b * bosonicProj a +
bosonicProj a * fermionicProj b - fermionicProj b * bosonicProj a +
fermionicProj a * bosonicProj b - bosonicProj b * fermionicProj a +
@ -779,12 +779,12 @@ lemma superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3
rw [h]
apply superCommuteF_grade h1 hs
lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule bosonic) :
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1)) := by
let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1))
@ -808,12 +808,12 @@ lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
simp_all [p, Finset.smul_sum]
· exact ha
lemma superCommuteF_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_fermionic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule fermionic) :
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1)) := by
let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1))
@ -870,6 +870,6 @@ lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
rw [← hn]
simpa using hc
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

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

@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.PerturbationTheory.FieldSpecification.TimeOrder
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.SuperCommute
import HepLean.PerturbationTheory.Koszul.KoszulSign
/-!
# Time Ordering in the CrAnAlgebra
# Time Ordering in the FieldOpFreeAlgebra
-/
@ -16,7 +16,7 @@ namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open FieldStatistic
namespace CrAnAlgebra
namespace FieldOpFreeAlgebra
noncomputable section
open HepLean.List
@ -26,35 +26,35 @@ open HepLean.List
-/
/-- Time ordering for the `CrAnAlgebra`. -/
def timeOrderF : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
/-- Time ordering for the `FieldOpFreeAlgebra`. -/
def timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs)
@[inherit_doc timeOrderF]
scoped[FieldSpecification.CrAnAlgebra] notation "𝓣ᶠ(" a ")" => timeOrderF a
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "𝓣ᶠ(" a ")" => timeOrderF a
lemma timeOrderF_ofCrAnList (φs : List 𝓕.CrAnStates) :
𝓣ᶠ(ofCrAnList φs) = crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs) := by
rw [← ofListBasis_eq_ofList]
simp only [timeOrderF, Basis.constr_basis]
lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
let pc (c : 𝓕.CrAnAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
let pc (c : 𝓕.FieldOpFreeAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c)
change pc c (Basis.mem_span _ c)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs, rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pc]
let pb (b : 𝓕.CrAnAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pb (b : 𝓕.FieldOpFreeAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓣ᶠ(a * b * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(b) * ofCrAnList φs)
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
· intro x hx
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.CrAnAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
let pa (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := 𝓣ᶠ(a * ofCrAnList φs' * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(ofCrAnList φs') * ofCrAnList φs)
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
@ -87,13 +87,13 @@ lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c)
· intro x hx h hp
simp_all [pc]
lemma timeOrderF_timeOrderF_right (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
lemma timeOrderF_timeOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
trans 𝓣ᶠ(a * b * 1)
· simp
· rw [timeOrderF_timeOrderF_mid]
simp
lemma timeOrderF_timeOrderF_left (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
lemma timeOrderF_timeOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
trans 𝓣ᶠ(1 * a * b)
· simp
· rw [timeOrderF_timeOrderF_mid]
@ -163,28 +163,28 @@ lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
simp_all
lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
rw [timeOrderF_timeOrderF_right,
timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
rw [timeOrderF_timeOrderF_left,
timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
rw [timeOrderF_timeOrderF_mid,
timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
@ -361,6 +361,6 @@ lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.
end
end CrAnAlgebra
end FieldOpFreeAlgebra
end FieldSpecification

2
HepLean/PerturbationTheory/FieldSpecification/CrAnSection.lean
View file

@ -22,7 +22,7 @@ That is to say, the states underlying `ψs` are the states in `φs`.
We denote these sections as `CrAnSection φs`.
Looking forward the main consequence of this definition is the lemma
`FieldSpecification.CrAnAlgebra.ofStateList_sum`.
`FieldSpecification.FieldOpFreeAlgebra.ofStateList_sum`.
In this module we define various properties of `CrAnSection`.

2
HepLean/PerturbationTheory/WicksTheorem.lean
View file

@ -20,7 +20,7 @@ Wick's theorem is related to Isserlis' theorem in mathematics.
namespace FieldSpecification
variable {𝓕 : FieldSpecification}
open CrAnAlgebra
open FieldOpFreeAlgebra
open FieldOpAlgebra
open HepLean.List
open WickContraction