Normal Order uncontracted some
FieldSpecification.WickAlgebra.normalOrder_uncontracted_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 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 is a consequence of definitions.
Exact Lean statement
lemma normalOrder_uncontracted_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) :
𝓝(ofFieldOpList [φsΛ ↩Λ φ i (some k)]ᵘᶜ)
= 𝓝(ofFieldOpList (optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedFieldOpEquiv φs φsΛ) k)))Formal artifact
Lean source
lemma normalOrder_uncontracted_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) : 𝓝(ofFieldOpList [φsΛ ↩Λ φ i (some k)]ᵘᶜ) = 𝓝(ofFieldOpList (optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedFieldOpEquiv φs φsΛ) k))) := by simp only [Nat.succ_eq_add_one, insertAndContract, optionEraseZ, uncontractedFieldOpEquiv, uncontractedListGet] congr rw [congr_uncontractedList] erw [uncontractedList_extractEquiv_symm_some] simp only [Fin.coe_succAboveEmb, List.map_eraseIdx, List.map_map] congr 1 conv_rhs => rw [get_eq_insertIdx_succAbove φ φs i]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/WickContractions.lean:89-100
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