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feat: Contractions and involutions

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jstoobysmith 2025-01-03 15:13:16 +00:00
parent bab9f10763
commit 7d1f15e18a
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36
HepLean/PerturbationTheory/Wick/CreateAnnihilateSection.lean
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@ -14,37 +14,37 @@ namespace Wick
open HepLean.List
open FieldStatistic
/-- The sections of `Σ i, f i` over a list `l : List 𝓕`.
/-- The sections of `Σ i, f i` over a list `φs : List 𝓕`.
In terms of physics, given some fields `φ₁...φₙ`, the different ways one can associate
each field as a `creation` or an `annilation` operator. E.g. the number of terms
`φ₁⁰φ₂¹...φₙ⁰` `φ₁¹φ₂¹...φₙ⁰` etc. If some fields are exclusively creation or annhilation
operators at this point (e.g. ansymptotic states) this is accounted for. -/
def CreateAnnihilateSect {𝓕 : Type} (f : 𝓕 → Type) (l : List 𝓕) : Type :=
Π i, f (l.get i)
def CreateAnnihilateSect {𝓕 : Type} (f : 𝓕 → Type) (φs : List 𝓕) : Type :=
Π i, f (φs.get i)
namespace CreateAnnihilateSect
section basic_defs
variable {𝓕 : Type} {f : 𝓕 → Type} [∀ i, Fintype (f i)] {l : List 𝓕} (a : CreateAnnihilateSect f l)
variable {𝓕 : Type} {f : 𝓕 → Type} [∀ i, Fintype (f i)] {φs : List 𝓕} (a : CreateAnnihilateSect f φs)
/-- The type `CreateAnnihilateSect f l` is finite. -/
instance fintype : Fintype (CreateAnnihilateSect f l) := Pi.fintype
/-- The type `CreateAnnihilateSect f φs` is finite. -/
instance fintype : Fintype (CreateAnnihilateSect f φs) := Pi.fintype
/-- The section got by dropping the first element of `l` if it exists. -/
def tail : {l : List 𝓕} → (a : CreateAnnihilateSect f l) → CreateAnnihilateSect f l.tail
/-- The section got by dropping the first element of `φs` if it exists. -/
def tail : {φs : List 𝓕} → (a : CreateAnnihilateSect f φs) → CreateAnnihilateSect f φs.tail
| [], a => a
| _ :: _, a => fun i => a (Fin.succ i)
/-- For a list of fields `i :: l` the value of the section at the head `i`. -/
def head {i : 𝓕} (a : CreateAnnihilateSect f (i :: l)) : f i := a ⟨0, Nat.zero_lt_succ l.length⟩
def head {φ : 𝓕} (a : CreateAnnihilateSect f (φ :: φs)) : f φ := a ⟨0, Nat.zero_lt_succ φs.length⟩
end basic_defs
section toList_basic
variable {𝓕 : Type} {f : 𝓕 → Type} (q : 𝓕 → FieldStatistic)
{l : List 𝓕} (a : CreateAnnihilateSect f l)
{φs : List 𝓕} (a : CreateAnnihilateSect f φs)
/-- The list `List (Σ i, f i)` defined by `a`. -/
def toList : {l : List 𝓕} → (a : CreateAnnihilateSect f l) → List (Σ i, f i)
@ -52,8 +52,8 @@ def toList : {l : List 𝓕} → (a : CreateAnnihilateSect f l) → List (Σ i,
| i :: _, a => ⟨i, a.head⟩ :: toList a.tail
@[simp]
lemma toList_length : (toList a).length = l.length := by
induction l with
lemma toList_length : (toList a).length = φs.length := by
induction φs with
| nil => rfl
| cons i l ih =>
simp only [toList, List.length_cons, Fin.zero_eta]
@ -64,13 +64,13 @@ lemma toList_tail : {l : List 𝓕} → (a : CreateAnnihilateSect f l) → toLis
| i :: l, a => by
simp [toList]
lemma toList_cons {i : 𝓕} (a : CreateAnnihilateSect f (i :: l)) :
lemma toList_cons {i : 𝓕} (a : CreateAnnihilateSect f (i :: φs)) :
(toList a) = ⟨i, a.head⟩ :: toList a.tail := by
rfl
lemma toList_get (a : CreateAnnihilateSect f l) :
(toList a).get = (fun i => ⟨l.get i, a i⟩) ∘ Fin.cast (by simp) := by
induction l with
lemma toList_get (a : CreateAnnihilateSect f φs) :
(toList a).get = (fun i => ⟨φs.get i, a i⟩) ∘ Fin.cast (by simp) := by
induction φs with
| nil =>
funext i
exact Fin.elim0 i
@ -89,8 +89,8 @@ lemma toList_get (a : CreateAnnihilateSect f l) :
@[simp]
lemma toList_grade :
FieldStatistic.ofList (fun i => q i.fst) a.toList = fermionic ↔
FieldStatistic.ofList q l = fermionic := by
induction l with
FieldStatistic.ofList q φs = fermionic := by
induction φs with
| nil =>
simp [toList]
| cons i r ih =>