Sign Insert None eq filterset
WickContraction.signInsertNone_eq_filterset
Plain-language statement
The following signs for a grading compliant Wick contraction are equal: - The sign φsΛ.signInsertNone φ φs i which is given by the following: For each contracted pair {a1, a2} in φsΛ if a1 < a2 such that i is within the range a1 < i < a2 we pick up a sign equal to 𝓢(φ, φs[a2]). - The sign got by moving φ through φ₀…φᵢ₋₁ and only picking...
Exact Lean statement
lemma signInsertNone_eq_filterset (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 = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter
(fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)⟩)Formal artifact
Lean source
lemma signInsertNone_eq_filterset (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 = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter (fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)⟩) := by rw [ofFinset_eq_prod, signInsertNone_eq_prod_getDual?_Some, map_prod] congr funext a simp only [Finset.mem_filter, Finset.mem_univ, true_and, Fin.getElem_fin] split · rename_i h simp only [h, true_and] split · rfl · simp only [map_one] · rename_i h simp [h] · exact hG- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertNone.lean:227-243
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