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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

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

Canonical source
Full Lean sourceLean 4
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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