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doc: Edits to Wick theorem docs

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jstoobysmith 2025-02-10 10:21:57 +00:00
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2
HepLean/PerturbationTheory/CreateAnnihilate.lean
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@ -11,7 +11,7 @@ import Mathlib.Algebra.BigOperators.Group.Finset
-/
/-- The type `CreateAnnihilate` is the type containing two elements `create` and `annihilate`.
This type is used to specify if an operator is a creation or annihilation operator
This type is used to specify if an operator is a creation, or annihilation, operator
or the sum thereof or intergral thereover etc. -/
inductive CreateAnnihilate where
| create : CreateAnnihilate

10
HepLean/PerturbationTheory/FeynmanDiagrams/Momentum.lean
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@ -48,7 +48,7 @@ We define the direct sum of the edge and vertex momentum spaces.
Corresponding to that spanned by its momentum. -/
def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
/-- The half momenta carries the structure of an addative commutative group. -/
/-- The half momenta carries the structure of an additive commutative group. -/
instance : AddCommGroup F.HalfEdgeMomenta := Pi.addCommGroup
/-- The half momenta carries the structure of a module over `ℝ`. Defined via its target. -/
@ -83,7 +83,7 @@ def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ]
Corresponding to that spanned by its total outflowing momentum. -/
def EdgeMomenta : Type := F.𝓔 → ℝ
/-- The edge momenta form an addative commuative group. -/
/-- The edge momenta form an additive commuative group. -/
instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
/-- The edge momenta form a module over `ℝ`. -/
@ -93,7 +93,7 @@ instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
Corresponding to that spanned by its total inflowing momentum. -/
def VertexMomenta : Type := F.𝓥 → ℝ
/-- The vertex momenta carries the structure of an addative commuative group. -/
/-- The vertex momenta carries the structure of an additive commuative group. -/
instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
/-- The vertex momenta carries the structure of a module over `ℝ`. -/
@ -106,7 +106,7 @@ def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
| 1 => F.VertexMomenta
/-- The target of the map `EdgeVertexMomentaMap` is either the type of edge momenta
or vertex momenta and thus carries the structure of an addative commuative group. -/
or vertex momenta and thus carries the structure of an additive commuative group. -/
instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
match i with
| 0 => instAddCommGroupEdgeMomenta F
@ -122,7 +122,7 @@ instance (i : Fin 2) : Module ℝ (EdgeVertexMomentaMap F i) :=
/-- The direct sum of `EdgeMomenta` and `VertexMomenta`. -/
def EdgeVertexMomenta : Type := DirectSum (Fin 2) (EdgeVertexMomentaMap F)
/-- The structure of a addative commutative group on `EdgeVertexMomenta` for a
/-- The structure of a additive commutative group on `EdgeVertexMomenta` for a
Feynman diagram `F`. -/
instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _

45
HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean
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@ -33,15 +33,18 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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 elements of the form
- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`, which (with the other conditions)
gives us that super-commutors are in the center.
- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` creation operators. I.e two creation
operators always super-commute.
- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` annihilation operators. I.e two
annihilation operators always super-commute.
- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` field operators with different statistics.
I.e. Fermions super-commute with bosons. -/
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 otherwords, 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-commutor is in the center of the
of the algebra.
-/
abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
namespace FieldOpAlgebra
@ -457,13 +460,15 @@ lemma ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero (x : FieldOpFreeAlgebra
-/
/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`, the element of
/-- 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`, the element of
/-- 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)
@ -487,7 +492,8 @@ 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`, the element of
/-- 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)
@ -500,7 +506,8 @@ lemma ofFieldOp_eq_sum (φ : 𝓕.FieldOp) :
simp only [map_sum]
rfl
/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`, the element of
/-- 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)
@ -528,8 +535,10 @@ remark notation_drop := "In doc-strings we will often drop explicit applications
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
annihilation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
If `φ` is an incoming asymptotic state this is zero by definition, otherwise
it is of the form `ofCrAnOp _`. -/
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
@ -552,8 +561,10 @@ lemma anPart_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
creation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
If `φ` is an outgoing asymptotic state this is zero by definition, otherwise
it is of the form `ofCrAnOp _`. -/
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

7
HepLean/PerturbationTheory/FieldOpAlgebra/Grading.lean
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@ -384,9 +384,10 @@ lemma directSum_eq_bosonic_plus_fermionic
abel
/-- For a field statistic `𝓕`, the algebra `𝓕.FieldOpAlgebra` is graded by `FieldStatistic`.
Those `ofCrAnList φs` for which `φs` has an overall `bosonic` statistic span `bosonic`
submodule, whilst those `ofCrAnList φs` for which `φs` has an overall `fermionic` statistic span
the `fermionic` submodule. -/
Those `ofCrAnList φs` for which `φs` has an overall `bosonic` statistic
(i.e. `𝓕 |>ₛ φs = bosonic`) span `bosonic`
submodule, whilst those `ofCrAnList φs` for which `φs` has an overall `fermionic` statistic
(i.e. `𝓕 |>ₛ φs = fermionic`) span the `fermionic` submodule. -/
instance fieldOpAlgebraGrade : GradedAlgebra (A := 𝓕.FieldOpAlgebra) statSubmodule where
one_mem := by
simp only [statSubmodule]

5
HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean
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@ -223,10 +223,11 @@ lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b)
`FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
defined as the decent of `ι ∘ₗ normalOrderF` from `FieldOpFreeAlgebra 𝓕` to `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`. -/
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

2
HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean
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@ -231,7 +231,7 @@ The proof of this result ultimately goes as follows
(considered as a list with one element) with
`ofCrAnList φsn` as a sum of supercommutors, one for each element of `φsn`.
- The fact that super-commutors are in the center of `𝓕.FieldOpAlgebra` is used to rearrange terms.
- Properties of ordered lists, and `normalOrderSign_eraseIdx` is then used to complete the proof.
- Properties of ordered lists, and `normalOrderSign_eraseIdx` are then used to complete the proof.
-/
lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)
(φs : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, 𝓝(ofCrAnList φs)]ₛ = ∑ n : Fin φs.length,

4
HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/WickContractions.lean
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@ -26,7 +26,7 @@ variable {𝓕 : FieldSpecification}
-/
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, and a `i ≤ φs.length` the following relation holds
`𝓕.FieldOp`, and a `i ≤ φs.length`, then the following relation holds:
`𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝(φ :: [φsΛ]ᵘᶜ)`
@ -101,7 +101,7 @@ lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp
/--
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`, then
`𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `FieldOp`
`𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `𝓕.FieldOp`
corresponding to `k` removed.
The proof of this result ultimately a consequence of definitions.

2
HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTerm.lean
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@ -68,7 +68,7 @@ lemma staticWickTerm_insert_zero_none (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
uncontracted fields in `φ₀…φₖ₋₁`
- the normal ordering `𝓝([φsΛ]ᵘᶜ.erase (uncontractedFieldOpEquiv φs φsΛ k))`.
The proof of this result ultimitley relies on
The proof of this result ultimately relies on
- `staticContract_insert_some` to rewrite static contractions.
- `normalOrder_uncontracted_some` to rewrite normal orderings.
- `sign_insert_some_zero` to rewrite signs.

3
HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean
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@ -371,7 +371,8 @@ lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
`FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
defined as the decent of `ι ∘ₗ timeOrderF` from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
defined as the decent of `ι ∘ₗ timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕` from
`FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
This decent exists because `ι ∘ₗ timeOrderF` is well-defined on equivalence classes.
The notation `𝓣(a)` is used for `timeOrder a`. -/

5
HepLean/PerturbationTheory/FieldOpAlgebra/Universality.lean
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@ -78,8 +78,9 @@ lemma universalLift_ι {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFie
For a field specification, `𝓕`, the algebra `𝓕.FieldOpAlgebra` satisfies the following universal
property. Let `f : 𝓕.CrAnFieldOp → A` be a function and `g : 𝓕.FieldOpFreeAlgebra →ₐ[ℂ] A`
the universal lift of that function associated with the free algebra `𝓕.FieldOpFreeAlgebra`.
If `g` is zero on the ideal defining `𝓕.FieldOpAlgebra`, then there is a unique
algebra map `g' : FieldOpAlgebra 𝓕 →ₐ[ℂ] A` such that `g' ∘ ι = g`.
If `g` is zero on the ideal defining `𝓕.FieldOpAlgebra`, then there exists
algebra map `g' : FieldOpAlgebra 𝓕 →ₐ[ℂ] A` such that `g' ∘ ι = g`, and furthermore this
algebra map is unique.
-/
lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp → A)
(h1 : ∀ a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet, FreeAlgebra.lift ℂ f a = 0) :

4
HepLean/PerturbationTheory/FieldOpAlgebra/WicksTheorem.lean
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@ -47,13 +47,13 @@ lemma wicks_theorem_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') :
rfl
remark wicks_theorem_context := "
In perturbation quantum field theory, Wick's theorem allows
In perturbative quantum field theory, Wick's theorem allows
us to expand expectation values of time-ordered products of fields in terms of normal-orders
and time contractions.
The theorem is used to simplify the calculation of scattering amplitudes, and is the precurser
to Feynman diagrams.
There is are actually three different versions of Wick's theorem used.
There are actually three different versions of Wick's theorem used.
The static version, the time-dependent version, and the normal-ordered time-dependent version.
HepLean contains a formalization of all three of these theorems in complete generality for
mixtures of bosonic and fermionic fields.

42
HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
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@ -41,13 +41,14 @@ abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ
namespace FieldOpFreeAlgebra
remark naming_convention := "
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` we will often postfix
their names with an `F` to indicate that they are related to the free algebra.
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` the postfix `F`
may be given to
their names to indicate that they are related to the free algebra.
This is to avoid confusion when working within the context of `FieldOpAlgebra` which is defined
as a quotient of `FieldOpFreeAlgebra`."
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
single `𝓕.CrAnFieldOp`. -/
/-- For a field specification `𝓕`, and a element `φ` of `𝓕.CrAnFieldOp`,
`ofCrAnOpF φ` is defined as the element of `𝓕.FieldOpFreeAlgebra` formed by `φ`. -/
def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
FreeAlgebra.ι ℂ φ
@ -70,9 +71,11 @@ lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp
ext x
simpa using congrFun hg x
/-- For a field specification `𝓕`, `ofCrAnListF φs` of `𝓕.FieldOpFreeAlgebra` formed by a
list `φs` of `𝓕.CrAnFieldOp`. For example for the list `[φ₁ᶜ, φ₂ᵃ, φ₃ᶜ]` we schematically
get `φ₁ᶜφ₂ᵃφ₃ᶜ`. The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.CrAnFieldOp`,
`ofCrAnListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
obtained by the product of `ofCrAnListF φ` for each `φ` in `φs`.
For example `ofCrAnListF [φ₁, φ₂, φ₃] = ofCrAnOpF φ₁ * ofCrAnOpF φ₂ * ofCrAnOpF φ₃`.
The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
@[simp]
@ -88,22 +91,25 @@ lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
`𝓕.FieldOp` by summing over the creation and annihilation components of `𝓕.FieldOp`.
For example for `φ₁` an incoming asymptotic field operator we get
`ofCrAnOpF φ₁ᶜ`, and for `φ₁` a
position field operator we get `ofCrAnOpF φ₁ᶜ + ofCrAnOpF φ₁ᵃ`. -/
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`,
`ofFieldOpF φ` is the element of `𝓕.FieldOpFreeAlgebra` formed by summing over
`ofCrAnOpF` of the
creation and annihilation parts of `φ`.
For example for `φ` an incoming asymptotic field operator we get
`ofCrAnOpF ⟨φ, ()⟩`, and for `φ` a
position field operator we get `ofCrAnOpF ⟨φ, .create⟩ + ofCrAnOpF ⟨φ, .annihilate⟩`. -/
def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
list of `𝓕.FieldOp` by summing over the creation and annihilation components.
For example, `φ₁` and `φ₂` position states `[φ1, φ2]` gets sent to
`(ofCrAnOpF φ1ᶜ + ofCrAnOpF φ1ᵃ) * (ofCrAnOpF φ2ᶜ + ofCrAnOpF φ2ᵃ)`. -/
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.FieldOp`,
`𝓕.ofFieldOpListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
obtained by the product of `ofFieldOpF φ` for each `φ` in `φs`.
For example `ofFieldOpListF [φ₁, φ₂, φ₃] = ofFieldOpF φ₁ * ofFieldOpF φ₂ * ofFieldOpF φ₃`. -/
def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
remark notation_drop := "In doc-strings we will often drop explicit applications of `ofCrAnOpF`,
`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF`"
remark notation_drop := "In doc-strings explicit applications of `ofCrAnOpF`,
`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF` will often be dropped."
/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩

6
HepLean/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean
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@ -237,8 +237,10 @@ lemma directSum_eq_bosonic_plus_fermionic
abel
/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpFreeAlgebra` is graded by `FieldStatistic`.
Those `ofCrAnListF φs` for which `φs` has an overall `bosonic` statistic span `bosonic`
submodule, whilst those `ofCrAnListF φs` for which `φs` has an overall `fermionic` statistic span
Those `ofCrAnListF φs` for which `φs` has an overall `bosonic` statistic
(i.e. `𝓕 |>ₛ φs = bosonic`) span `bosonic`
submodule, whilst those `ofCrAnListF φs` for which `φs` has an overall `fermionic` statistic
(i.e. `𝓕 |>ₛ φs = fermionic`) span
the `fermionic` submodule. -/
instance fieldOpFreeAlgebraGrade :
GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where

5
HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
View file

@ -31,8 +31,11 @@ noncomputable section
`FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕`
defined by its action on the basis `ofCrAnListF φs`, taking `ofCrAnListF φs` to
`normalOrderSign φs • ofCrAnListF (normalOrderList φs)`.
That is, `normalOrderF` normal-orders the field operators and multiplies by the sign of the
normal order.
The notation `𝓝ᶠ(a)` is used for `normalOrderF a`. -/
The notation `𝓝ᶠ(a)` is used for `normalOrderF a` for `a` an element of
`FieldOpFreeAlgebra 𝓕`. -/
def normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=
Basis.constr ofCrAnListFBasis ℂ fun φs =>
normalOrderSign φs • ofCrAnListF (normalOrderList φs)

2
HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
View file

@ -405,7 +405,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : L
simp [mul_comm]
/--
For a field specification `𝓕`, and to lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
For a field specification `𝓕`, and two lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
the following super commutation relation holds:
`[φs', φ₀…φₙ]ₛca = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛca * φᵢ₊₁ … φₙ`

2
HepLean/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
View file

@ -30,6 +30,8 @@ open HepLean.List
`FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕`
defined by its action on the basis `ofCrAnListF φs`, taking
`ofCrAnListF φs` to `crAnTimeOrderSign φs • ofCrAnListF (crAnTimeOrderList φs)`.
That is, `timeOrderF` time-orders the field operators and multiplies by the sign of the
time order.
The notation `𝓣ᶠ(a)` is used for `timeOrderF a` -/
def timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 :=

42
HepLean/PerturbationTheory/FieldSpecification/Basic.lean
View file

@ -42,13 +42,15 @@ The structure `FieldSpecification` is defined to have the following content:
- For `f` a *Weyl fermion*, `PositionLabel f` will have four elements, one for each Lorentz
index of the field and its conjugate.
- For every field `f` in `Field`, a type `AsymptoticLabel f` whose elements label the different
asymptotic based field operators associated with the field `f`. For example,
types of incoming asymptotic field operators associated with the
field `f` (this is also matches the types of outgoing asymptotic field operators).
For example,
- For `f` a *real-scalar field*, `AsymptoticLabel f` will have a unique element.
- For `f` a *complex-scalar field*, `AsymptoticLabel f` will have two elements, one for the
field operator and one for its conjugate.
- For `f` a *Dirac fermion*, `AsymptoticLabel f` will have four elements, two for each spin.
- For `f` a *Weyl fermion*, `AsymptoticLabel f` will have two elements, one for each spin.
- For each field `f` in `Field`, a field statistic `statistic f` which classifying `f` as either
- For each field `f` in `Field`, a field statistic `statistic f` which classifies `f` as either
`bosonic` or `fermionic`.
-/
structure FieldSpecification where
@ -67,20 +69,30 @@ structure FieldSpecification where
namespace FieldSpecification
variable (𝓕 : FieldSpecification)
/-- For a field specification `𝓕`, the type `𝓕.FieldOp` is defined such that every element of
`FieldOp` corresponds either to:
- an incoming asymptotic field operator `.inAsymp` which is specified by
a field `f` in `𝓕.Field`, an element of `AsymptoticLabel f` (which specifies exactly
which asymptotic field operator associated with `f`) and a `3`-momentum.
- an position operator `.position` which is specified by
a field `f` in `𝓕.Field`, an element of `PositionLabel f` (which specifies exactly
which position field operator associated with `f`) and a element of `SpaceTime`.
- an outgoing asymptotic field operator `.outAsymp` which is specified by
a field `f` in `𝓕.Field`, an element of `AsymptoticLabel f` (which specifies exactly
which asymptotic field operator associated with `f`) and a `3`-momentum.
/-- For a field specification `𝓕`, the inductive type `𝓕.FieldOp` is defined
to contain the following elements:
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
element labelled `inAsymp f e p` corresponding to an incoming asymptotic field operator
of the field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
- for every `f` in `𝓕.Field`, element of `e` of `PositionLabel f` and space-time position `x`, an
element labelled `position f e x` corresponding to a position field operator of the field `f`,
of label `e` (e.g. specifying the Lorentz index), and position `x`.
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
element labelled `outAsymp f e p` corresponding to an outgoing asymptotic field operator of the
field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
Note the use of `3`-momentum here rather then `4`-momentum. This is because the asymptotic states
have on-shell momenta.
As some intuition, if `f` corresponds to a Weyl-fermion field, then
- For `inAsymp f e p`, `e` would correspond to a spin `s`, and `inAsymp f e p` would, once
represented in the operator algebra,
be proportional to the creation operator `a(p, s)`.
- `position f e x`, `e` would correspond to a Lorentz index `α`, and `position f e x` would,
once represented in the operator algebra, be proportional to the operator
`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x} + y_α(p,s) a^†(p, s) e^{-i p x})`.
- `outAsymp f e p`, `e` would corresond to a spin `s`, and `outAsymp f e p` would,
once represented in the operator algebra, be proportional to the
annihilation operator `a^†(p, s)`.
This type contains all operators which are related to a field.
-/
inductive FieldOp (𝓕 : FieldSpecification) where
| inAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp

61
HepLean/PerturbationTheory/FieldSpecification/CrAnFieldOp.lean
View file

@ -63,15 +63,31 @@ def fieldOpToCreateAnnihilateTypeCongr : {i j : 𝓕.FieldOp} → i = j →
| _, _, rfl => Equiv.refl _
/--
For a field specification `𝓕`, elements in `𝓕.CrAnFieldOp`, the type
of creation and annihilation field operators, corresponds to
- an incoming asymptotic field operator `.inAsymp` in `𝓕.FieldOp`.
- a position operator `.position` in `𝓕.FieldOp` and an element of
`CreateAnnihilate` specifying the creation or annihilation part of that position operator.
- an outgoing asymptotic field operator `.outAsymp` in `𝓕.FieldOp`.
For a field specification `𝓕`, the (sigma) type `𝓕.CrAnFieldOp`
corresponds to the type of creation and annihilation parts of field operators.
It formally defined to consist of the following elements:
- for each in incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the creation part of `φ`.
Here `φ` has no annihilation part. (Here `()` is the unique element of `Unit`.)
- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
written as `⟨φ, .create⟩`, corresponding to the creation part of `φ`.
- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
written as `⟨φ, .annihilate⟩`, corresponding to the annihilation part of `φ`.
- for each out outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the annihilation part of `φ`.
Here `φ` has no creation part. (Here `()` is the unique element of `Unit`.)
As some intuition, if `f` corresponds to a Weyl-fermion field, it would contribute
the following elements to `𝓕.CrAnFieldOp`
- an element corresponding to incoming asymptotic operators for each spin `s`: `a(p, s)`.
- an element corresponding to the creation parts of position operators for each each Lorentz
index `α`:
`∑ s, ∫ d^3p/(…) (x_α(p,s) a(p, s) e^{-i p x})`.
- an element corresponding to anihilation parts of position operator,
for each each Lorentz index `α`:
`∑ s, ∫ d^3p/(…) (y_α(p,s) a^†(p, s) e^{-i p x})`.
- an element corresponding to outgoing asymptotic operators for each spin `s`: `a^†(p, s)`.
Note that the incoming and outgoing asymptotic field operators are implicitly creation and
annihilation operators respectively.
-/
def CrAnFieldOp : Type := Σ (s : 𝓕.FieldOp), 𝓕.fieldOpToCrAnType s
@ -82,8 +98,13 @@ def crAnFieldOpToFieldOp : 𝓕.CrAnFieldOp → 𝓕.FieldOp := Sigma.fst
lemma crAnFieldOpToFieldOp_prod (s : 𝓕.FieldOp) (t : 𝓕.fieldOpToCrAnType s) :
𝓕.crAnFieldOpToFieldOp ⟨s, t⟩ = s := rfl
/-- The map from creation and annihilation states to the type `CreateAnnihilate`
specifying if a state is a creation or an annihilation state. -/
/-- For a field specficiation `𝓕`, `𝓕.crAnFieldOpToCreateAnnihilate` is the map from
`𝓕.CrAnFieldOp` to `CreateAnnihilate` taking `φ` to `create` if
- `φ` corresponds to an incoming asymptotic field operator or the creation part of a position based
field operator.
otherwise it takes `φ` to `annihilate`.
-/
def crAnFieldOpToCreateAnnihilate : 𝓕.CrAnFieldOp → CreateAnnihilate
| ⟨FieldOp.inAsymp _, _⟩ => CreateAnnihilate.create
| ⟨FieldOp.position _, CreateAnnihilate.create⟩ => CreateAnnihilate.create
@ -92,7 +113,7 @@ def crAnFieldOpToCreateAnnihilate : 𝓕.CrAnFieldOp → CreateAnnihilate
/-- For a field specification `𝓕`, and an element `φ` in `𝓕.CrAnFieldOp`, the field
statistic `crAnStatistics φ` is defined to be the statistic associated with the field `𝓕.Field`
(or `𝓕.FieldOp`) underlying `φ`.
(or the `𝓕.FieldOp`) underlying `φ`.
The following notation is used in relation to `crAnStatistics`:
- For `φ` an element of `𝓕.CrAnFieldOp`, `𝓕 |>ₛ φ` is `crAnStatistics φ`.
@ -115,18 +136,18 @@ scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
scoped[FieldSpecification] infixl:80 "|>ᶜ" =>
crAnFieldOpToCreateAnnihilate
remark notation_remark := "When working with a field specification `𝓕` we will use
some notation within doc-strings and in code. The main notation used is:
- In doc-strings when field statistics occur in exchange signs we may drop the `𝓕 |>ₛ _`.
- In doc-strings we will often write lists of `FieldOp` or `CrAnFieldOp` `φs` as e.g. `φ₀…φₙ`,
remark notation_remark := "When working with a field specification `𝓕` the
following notation will be used within doc-strings:
- when field statistics occur in exchange signs the `𝓕 |>ₛ _` may be dropped.
- lists of `FieldOp` or `CrAnFieldOp` `φs` may be written as `φ₀…φₙ`,
which should be interpreted within the context in which it appears.
- In doc-strings we may use e.g. `φᶜ` to indicate the creation part of an operator and
- `φᶜ` may be used to indicate the creation part of an operator and
`φᵃ` to indicate the annihilation part of an operator.
Some examples:
- `𝓢(φ, φs)` corresponds to `𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs)`
- `φ₀…φᵢ₋₁φᵢ₊₁…φₙ` corresponds to a (given) list `φs = φ₀…φₙ` with the element at the `i`th position
removed.
Some examples of these notation-conventions are:
- `𝓢(φ, φs)` which corresponds to `𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs)`
- `φ₀…φᵢ₋₁φᵢ₊₁…φₙ` which corresponds to a (given) list `φs = φ₀…φₙ` with the element at the
`i`th position removed.
"
end FieldSpecification

9
HepLean/PerturbationTheory/FieldSpecification/NormalOrder.lean
View file

@ -17,8 +17,8 @@ variable {𝓕 : FieldSpecification}
/-- For a field specification `𝓕`, `𝓕.normalOrderRel` is a relation on `𝓕.CrAnFieldOp`
representing normal ordering. It is defined such that `𝓕.normalOrderRel φ₀ φ₁`
is true if one of the following is true
- `φ₀` is a creation operator
- `φ₁` is an annihilation.
- `φ₀` is a field creation operator
- `φ₁` is a field annihilation operator.
Thus, colloquially `𝓕.normalOrderRel φ₀ φ₁` says the creation operators are 'less then'
annihilation operators. -/
@ -344,11 +344,12 @@ lemma normalOrderList_eraseIdx_normalOrderEquiv {φs : List 𝓕.CrAnFieldOp} (n
rw [HepLean.List.eraseIdx_insertionSort_fin]
/-- For a field specification `𝓕`, a list `φs = φ₀…φₙ` of `𝓕.CrAnFieldOp` and an `i < φs.length`,
the following relation holds
then
`normalOrderSign (φ₀…φᵢ₋₁φᵢ₊₁…φₙ)` is equal to the product of
- `normalOrderSign φ₀…φₙ`,
- `𝓢(φᵢ, φ₀…φᵢ₋₁)` i.e. the sign needed to remove `φᵢ` from `φ₀…φₙ`,
- `𝓢(φᵢ, _)` where `_` is the list of elements appearing before `φᵢ` after normal ordering. I.e.
- `𝓢(φᵢ, _)` where `_` is the list of elements appearing before `φᵢ` after normal ordering,
i.e.
the sign needed to insert `φᵢ` back into the normal-ordered list at the correct place. -/
lemma normalOrderSign_eraseIdx (φs : List 𝓕.CrAnFieldOp) (i : Fin φs.length) :
normalOrderSign (φs.eraseIdx i) = normalOrderSign φs *

4
HepLean/PerturbationTheory/FieldSpecification/TimeOrder.lean
View file

@ -193,7 +193,7 @@ lemma timeOrderList_eq_maxTimeField_timeOrderList (φ : 𝓕.FieldOp) (φs : Lis
/-- For a field specification `𝓕`, `𝓕.crAnTimeOrderRel` is a relation on
`𝓕.CrAnFieldOp` representing time ordering.
It is defined as such that `𝓕.crAnTimeOrderRel φ₀ φ₁` is true if and only if one of the following
It is defined such that `𝓕.crAnTimeOrderRel φ₀ φ₁` is true if and only if one of the following
holds
- `φ₀` is an *outgoing* asymptotic operator
- `φ₁` is an *incoming* asymptotic field operator
@ -201,6 +201,8 @@ lemma timeOrderList_eq_maxTimeField_timeOrderList (φ : 𝓕.FieldOp) (φs : Lis
the `SpaceTime` point of `φ₀` has a time *greater* then or equal to that of `φ₁`.
Thus, colloquially `𝓕.crAnTimeOrderRel φ₀ φ₁` if `φ₀` has time *greater* then or equal to `φ₁`.
The use of *greater* then rather then *less* then is because on ordering lists of operators
it is needed that the operator with the greatest time is to the left.
-/
def crAnTimeOrderRel (a b : 𝓕.CrAnFieldOp) : Prop := 𝓕.timeOrderRel a.1 b.1

4
HepLean/PerturbationTheory/FieldStatistics/ExchangeSign.lean
View file

@ -27,8 +27,8 @@ variable {𝓕 : Type}
/-- The exchange sign, `exchangeSign`, is defined as the group homomorphism
`FieldStatistic →* FieldStatistic →* ℂ`,
for which `exchangeSign a b` is `-1` if both `a` and `b` are `fermionic` and `1` otherwise.
The exchange sign is sign one picks up on exchanging an operator or field `φ₁` of statistic `a`
with one `φ₂` of statistic `b`, i.e. `φ₁φ₂ → φ₂φ₁`.
The exchange sign is the sign one picks up on exchanging an operator or field `φ₁` of statistic
`a` with an operator or field `φ₂` of statistic `b`, i.e. `φ₁φ₂ → φ₂φ₁`.
The notation `𝓢(a, b)` is used for the exchange sign of `a` and `b`. -/
def exchangeSign : FieldStatistic →* FieldStatistic →* ℂ where

18
HepLean/PerturbationTheory/WickContraction/Basic.lean
View file

@ -19,20 +19,6 @@ contracting, a Wick contraction
is a finite set of pairs of `Fin n` (numbers `0`, …, `n-1`), such that no
element of `Fin n` occurs in more then one pair. The pairs are the positions of fields we
'contract' together.
For example for `n = 3` there are `4` Wick contractions:
- `∅`, corresponding to the case where no fields are contracted.
- `{{0, 1}}`, corresponding to the case where the field at position `0` and `1` are contracted.
- `{{0, 2}}`, corresponding to the case where the field at position `0` and `2` are contracted.
- `{{1, 2}}`, corresponding to the case where the field at position `1` and `2` are contracted.
For `n=4` some possible Wick contractions are
- `∅`, corresponding to the case where no fields are contracted.
- `{{0, 1}, {2, 3}}`, corresponding to the case where the field at position `0` and `1` are
contracted and the field at position `2` and `3` are contracted.
- `{{0, 2}, {1, 3}}`, corresponding to the case where the field at position `0` and `2` are
contracted and the field at position `1` and `3` are contracted.
etc.
-/
def WickContraction (n : ℕ) : Type :=
{f : Finset ((Finset (Fin n))) // (∀ a ∈ f, a.card = 2) ∧
@ -534,8 +520,8 @@ lemma prod_finset_eq_mul_fst_snd (c : WickContraction n) (a : c.1)
/-- For a field specification `𝓕`, `φs` a list of `𝓕.FieldOp` and a Wick contraction
`φsΛ` of `φs`, the Wick contraction `φsΛ` is said to be `GradingCompliant` if
for every pair in `φsΛ` the contracted fields are either both `fermionic` or both `bosonic`.
I.e. in a `GradingCompliant` Wick contraction no contractions occur between `fermionic` and
`bosonic` fields. -/
In other words, in a `GradingCompliant` Wick contraction no contractions occur between
`fermionic` and `bosonic` fields. -/
def GradingCompliant (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :=
∀ (a : φsΛ.1), (𝓕 |>ₛ φs[φsΛ.fstFieldOfContract a]) = (𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a])

48
HepLean/PerturbationTheory/WickContraction/ExtractEquiv.lean
View file

@ -107,4 +107,52 @@ lemma sum_extractEquiv_congr [AddCommMonoid M] {n m : ℕ} (i : Fin n) (f : Wick
rw [Finset.sum_sigma']
rfl
/-- For `n = 3` there are `4` possible Wick contractions:
- `∅`, corresponding to the case where no fields are contracted.
- `{{0, 1}}`, corresponding to the case where the field at position `0` and `1` are contracted.
- `{{0, 2}}`, corresponding to the case where the field at position `0` and `2` are contracted.
- `{{1, 2}}`, corresponding to the case where the field at position `1` and `2` are contracted.
The proof of this result uses the fact that Lean is an executable programming language
and can calculate all Wick contractions for a given `n`. -/
lemma mem_three (c : WickContraction 3) : c.1 ∈ ({∅, {{0, 1}}, {{0, 2}}, {{1, 2}}} :
Finset (Finset (Finset (Fin 3)))) := by
fin_cases c <;>
simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Function.Embedding.coeFn_mk,
Finset.mem_insert, Finset.mem_singleton]
· exact Or.inl rfl
· exact Or.inr (Or.inl rfl)
· exact Or.inr (Or.inr (Or.inl rfl))
· exact Or.inr (Or.inr (Or.inr rfl))
/-- For `n = 4` there are `10` possible Wick contractions including e.g.
- `∅`, corresponding to the case where no fields are contracted.
- `{{0, 1}, {2, 3}}`, corresponding to the case where the fields at position `0` and `1` are
contracted, and the fields at position `2` and `3` are contracted.
- `{{0, 2}, {1, 3}}`, corresponding to the case where the fields at position `0` and `2` are
contracted, and the fields at position `1` and `3` are contracted.
The proof of this result uses the fact that Lean is an executable programming language
and can calculate all Wick contractions for a given `n`.
-/
lemma mem_four (c : WickContraction 4) : c.1 ∈ ({∅,
{{0, 1}}, {{0, 2}}, {{0, 3}}, {{1, 2}}, {{1, 3}}, {{2,3}},
{{0, 1}, {2, 3}}, {{0, 2}, {1, 3}}, {{0, 3}, {1, 2}}} :
Finset (Finset (Finset (Fin 4)))) := by
fin_cases c <;>
simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Function.Embedding.coeFn_mk,
Finset.mem_insert, Finset.mem_singleton]
· exact Or.inl rfl -- ∅
· exact Or.inr (Or.inl rfl) -- {{0, 1}}
· exact Or.inr (Or.inr (Or.inl rfl)) -- {{0, 2}}
· exact Or.inr (Or.inr (Or.inr (Or.inl rfl))) -- {{0, 3}}
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))) -- {{1, 2}}
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr rfl))))))))
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl))))) -- {{1, 3}}
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl))))))))
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))))) -- {{2, 3 }}
· exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl rfl)))))))
end WickContraction

8
HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean
View file

@ -25,7 +25,7 @@ open HepLean.Fin
-/
/-- Given a Wick contraction `φsΛ` for a list `φs` of `𝓕.FieldOp`,
a `𝓕.FieldOp` `φ`, an `i ≤ φs.length` and a `j` which is either `none` or
an element `φ` of `𝓕.FieldOp`, an `i ≤ φs.length` and a `j` which is either `none` or
some element of `φsΛ.uncontracted`, the new Wick contraction
`φsΛ.insertAndContract φ i j` is defined by inserting `φ` into `φs` after
the first `i`-elements and moving the values representing the contracted pairs in `φsΛ`
@ -35,6 +35,8 @@ open HepLean.Fin
In other words, `φsΛ.insertAndContract φ i j` is formed by adding `φ` to `φs` at position `i`,
and contracting `φ` with the field originally at position `j` if `j` is not none.
It is a Wick contraction of `φs.insertIdx φ i`, the list `φs` with `φ` inserted at
position `i`.
The notation `φsΛ ↩Λ φ i j` is used to denote `φsΛ.insertAndContract φ i j`. -/
def insertAndContract {φs : List 𝓕.FieldOp} (φ : 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
@ -285,11 +287,11 @@ lemma insert_fin_eq_self (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}
rfl
/-- For a list `φs` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, a `i ≤ φs.length` a sum over
`𝓕.FieldOp` and a `i ≤ φs.length` then a sum over
Wick contractions of `φs` with `φ` inserted at `i` is equal to the sum over Wick contractions
`φsΛ` of just `φs` and the sum over optional uncontracted elements of the `φsΛ`.
I.e. `∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ` is equal to
In other words, `∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ` is equal to
`∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) `.
where `(φs.insertIdx i φ)` is `φs` with `φ` inserted at position `i`. -/
lemma insertLift_sum (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}

8
HepLean/PerturbationTheory/WickContraction/Join.lean
View file

@ -25,7 +25,13 @@ open FieldOpAlgebra
/-- Given a list `φs` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs` and a Wick contraction
`φsucΛ` of `[φsΛ]ᵘᶜ`, `join φsΛ φsucΛ` is defined as the Wick contraction of `φs` consisting of
the contractions in `φsΛ` and those in `φsucΛ`. -/
the contractions in `φsΛ` and those in `φsucΛ`.
As an example, for `φs = [φ1, φ2, φ3, φ4]`,
`φsΛ = {{0, 1}}` corresponding to the contraction of `φ1` and `φ2` in `φs` and
`φsucΛ = {{0, 1}}`
corresponding to the contraction of `φ3` and `φ4` in `[φsΛ]ᵘᶜ = [φ3, φ4]`, then
`join φsΛ φsucΛ` is the contraction `{{0, 1}, {2, 3}}` of `φs`. -/
def join {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) : WickContraction φs.length :=
⟨φsΛ.1 ∪ φsucΛ.1.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding, by

5
HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
View file

@ -30,8 +30,9 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
/-- For a list `φs` of `𝓕.FieldOp`, and a Wick contraction `φsΛ` of `φs`,
the complex number `φsΛ.sign` is defined to be the sign (`1` or `-1`) corresponding
to the number of `fermionic`-`fermionic` exchanges that must done to put
contracted pairs with `φsΛ` next to one another, starting from the contracted pair
to the number of `fermionic`-`fermionic` exchanges that must be done to put
contracted pairs within `φsΛ` next to one another, starting recursively
from the contracted pair
whose first element occurs at the left-most position. -/
def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],

4
HepLean/PerturbationTheory/WickContraction/Sign/InsertNone.lean
View file

@ -239,7 +239,7 @@ lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
· exact hG
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
an `i ≤ φs.length` and a `φ` in `𝓕.FieldOp`, the following relation holds
an `i ≤ φs.length`, and a `φ` in `𝓕.FieldOp`, then
`(φsΛ ↩Λ φ i none).sign = s * φsΛ.sign`
where `s` is the sign got by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
are contracted with some element.
@ -255,7 +255,7 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
exact hG
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
and a `φ` in `𝓕.FieldOp`, the following relation holds `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
and a `φ` in `𝓕.FieldOp`, then `(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign`.
This is a direct corollary of `sign_insert_none`. -/
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)

2
HepLean/PerturbationTheory/WickContraction/Sign/Join.lean
View file

@ -422,7 +422,7 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
`(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
In `φsΛ.sign` the sign is determined by starting with the contracted pair
whose first element occurs at the left-most position. This lemma manifests that
whose first element occurs at the left-most position. This lemma manifests that this
choice does not matter, and that contracted pairs can be brought together in any order. -/
lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :

6
HepLean/PerturbationTheory/WickContraction/StaticContract.lean
View file

@ -19,9 +19,9 @@ variable {n : ℕ} (c : WickContraction n)
open HepLean.List
open FieldOpAlgebra
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
/-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ`, the
element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.staticContract` is defined as the product
of `[anPart φs[j], φs[k]]ₛ` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
of `[anPart φs[j], φs[k]]ₛ` over contracted pairs `{j, k}` in `φsΛ`
with `j < k`. -/
noncomputable def staticContract {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) :
@ -31,7 +31,7 @@ noncomputable def staticContract {φs : List 𝓕.FieldOp}
superCommute_anPart_ofFieldOp_mem_center _ _⟩
/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, and a `i ≤ φs.length` the following relation holds
`𝓕.FieldOp`, and a `i ≤ φs.length`, then the following relation holds:
`(φsΛ ↩Λ φ i none).staticContract = φsΛ.staticContract`