Static Contract insert some
WickContraction.staticContract_insert_some
Plain-language statement
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 (φsΛ ↩Λ φ i (some k)).staticContract is equal to the product of - [anPart φ, φs[k]]ₛ if i ≤ k or [anPart φs[k], φ]ₛ if k < i - φsΛ.staticContract. The proof of this result ultimately...
Exact Lean statement
lemma staticContract_insert_some
(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
(φsΛ ↩Λ φ i (some j)).staticContract =
(if i < i.succAbove j then
⟨[anPart φ, ofFieldOp φs[j.1]]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩
else ⟨[anPart φs[j.1], ofFieldOp φ]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩) *
φsΛ.staticContractFormal artifact
Lean source
lemma staticContract_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) : (φsΛ ↩Λ φ i (some j)).staticContract = (if i < i.succAbove j then ⟨[anPart φ, ofFieldOp φs[j.1]]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩ else ⟨[anPart φs[j.1], ofFieldOp φ]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩) * φsΛ.staticContract := by rw [staticContract, insertAndContract_some_prod_contractions] congr 1 · simp only [Nat.succ_eq_add_one, insertAndContract_fstFieldOfContract_some_incl, finCongr_apply, List.get_eq_getElem, insertAndContract_sndFieldOfContract_some_incl, Fin.getElem_fin] split <;> simp · congr with a simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/PerturbationTheory/WickContraction/StaticContract.lean:61-75
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