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docs: More docs related to Wicks theorem

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jstoobysmith 2025-02-05 11:52:55 +00:00
parent 759f204ed5
commit 8434334bbf
10 changed files with 124 additions and 61 deletions
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10
HepLean/PerturbationTheory/WickContraction/Sign/InsertNone.lean
View file

@ -157,11 +157,6 @@ lemma signInsertNone_eq_prod_prod (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
erw [hG a]
rfl
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) : (φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
rw [sign_insert_none_eq_signInsertNone_mul_sign]
simp [signInsertNone]
lemma signInsertNone_eq_prod_getDual?_Some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
φsΛ.signInsertNone φ φs i = ∏ (x : Fin φs.length),
@ -255,4 +250,9 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
rw [signInsertNone_eq_filterset]
exact hG
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) : (φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
rw [sign_insert_none_eq_signInsertNone_mul_sign]
simp [signInsertNone]
end WickContraction

11
HepLean/PerturbationTheory/WickContraction/Sign/InsertSome.lean
View file

@ -901,4 +901,15 @@ lemma sign_insert_some_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
rw [mul_comm, ← mul_assoc]
simp
lemma sign_insert_some_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted)
(hn : GradingCompliant φs φsΛ ∧ (𝓕|>ₛφ) = 𝓕|>ₛφs[k.1]):
(φsΛ ↩Λ φ 0 k).sign = 𝓢(𝓕|>ₛφ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x < ↑k))⟩) *
φsΛ.sign := by
rw [sign_insert_some_of_not_lt]
· simp
· simp
· exact hn
end WickContraction

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

@ -417,6 +417,11 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
apply sign_congr
exact join_uncontractedListGet (singleton hij) φsucΛ'
/-- Let `φsΛ` be a grading compliant Wick contraction for `φs` and
`φsucΛ` a Wick contraction for `[φsΛ]ᵘᶜ`. Then `(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign`.
This lemma manifests the fact that it does not matter which order contracted pairs are brought
together when defining the sign of a contraction. -/
lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :
(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign := by