Normal Order F swap create annihilate of Cr An List F of Cr An List F
FieldSpecification.FieldOpFreeAlgebra.normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF
Plain-language statement
For a field specification 𝓕, normalOrderF is the linear map FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 defined by its action on the basis ofCrAnListF φs, taking ofCrAnListF φs to normalOrderSign φs • ofCrAnListF (normalOrderList φs). That is, normalOrderF normal-orders the field operators and multiplies by the sign of the normal orde...
Exact Lean statement
lemma normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(φs φs' : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF φs) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs)Formal artifact
Lean source
lemma normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF (φc φa : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate) (φs φs' : List 𝓕.CrAnFieldOp) : 𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φc * ofCrAnOpF φa * ofCrAnListF φs) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) • 𝓝ᶠ(ofCrAnListF φs' * ofCrAnOpF φa * ofCrAnOpF φc * ofCrAnListF φs) := by rw [mul_assoc, mul_assoc, ← ofCrAnListF_cons, ← ofCrAnListF_cons, ← ofCrAnListF_append] rw [normalOrderF_ofCrAnListF, normalOrderSign_swap_create_annihilate φc φa hφc hφa] rw [normalOrderList_swap_create_annihilate φc φa hφc hφa, ← smul_smul, ← normalOrderF_ofCrAnListF] rw [ofCrAnListF_append, ofCrAnListF_cons, ofCrAnListF_cons] noncomm_ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean:186-195
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