Time Order of Field Op List eq Time Only
FieldSpecification.WickAlgebra.timeOrder_ofFieldOpList_eqTimeOnly
Plain-language statement
For a list φs of 𝓕.FieldOp, then 𝓣(φs) = ∑ φsΛ, φsΛ.sign • φsΛ.timeContract * 𝓣(𝓝([φsΛ]ᵘᶜ)) where the sum is over all Wick contraction φsΛ which only have equal time contractions. This result follows from - static_wick_theorem to rewrite 𝓣(φs) on the left hand side as a sum of 𝓣(φsΛ.staticWickTerm). - `EqTimeOnly.timeOrder_staticContra...
Exact Lean statement
lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.FieldOp) :
𝓣(ofFieldOpList φs) = ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs)}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))Formal artifact
Lean source
lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.FieldOp) : 𝓣(ofFieldOpList φs) = ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs)}), φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by rw [static_wick_theorem φs] let e2 : WickContraction φs.length ≃ {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} ⊕ {φsΛ : WickContraction φs.length // ¬ φsΛ.EqTimeOnly} := (Equiv.sumCompl _).symm rw [← e2.symm.sum_comp] simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type, Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, map_add, map_sum, e2] simp only [staticWickTerm, Algebra.smul_mul_assoc, map_smul] conv_lhs => enter [2, 2, x] rw [timeOrder_timeOrder_left, timeOrder_staticContract_of_not_mem _ x.2] simp only [zero_mul, map_zero, smul_zero, Finset.sum_const_zero, add_zero] congr funext x rw [staticContract_eq_timeContract_of_eqTimeOnly, timeOrder_timeContract_mul_of_eqTimeOnly_left] <;> exact x.2- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/PerturbationTheory/WickAlgebra/WicksTheoremNormal.lean:46-66
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