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Project-declaredLean 4.32.0 ยท mathlib@81a5d257c8e4

Super Commute F of Cr An List F of Cr An List F cons

FieldSpecification.FieldOpFreeAlgebra.superCommuteF_ofCrAnListF_ofCrAnListF_cons

physicsquantum field theoryrelativitysuper commute

Plain-language statement

For a field specification ๐“•, the super commutator superCommuteF is defined as the linear map ๐“•.FieldOpFreeAlgebra โ†’โ‚—[โ„‚] ๐“•.FieldOpFreeAlgebra โ†’โ‚—[โ„‚] ๐“•.FieldOpFreeAlgebra which on the lists ฯ†s and ฯ†s' of ๐“•.CrAnFieldOp gives superCommuteF ฯ†s ฯ†s' = ฯ†s * ฯ†s' - ๐“ข(ฯ†s, ฯ†s') โ€ข ฯ†s' * ฯ†s. The notation [a, b]โ‚›F can be used for superCommuteF a b...

Exact Lean statement

lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (ฯ† : ๐“•.CrAnFieldOp) (ฯ†s ฯ†s' : List ๐“•.CrAnFieldOp) :
    [ofCrAnListF ฯ†s, ofCrAnListF (ฯ† :: ฯ†s')]โ‚›F =
    [ofCrAnListF ฯ†s, ofCrAnOpF ฯ†]โ‚›F * ofCrAnListF ฯ†s' +
    ๐“ข(๐“• |>โ‚› ฯ†s, ๐“• |>โ‚› ฯ†)
    โ€ข ofCrAnOpF ฯ† * [ofCrAnListF ฯ†s, ofCrAnListF ฯ†s']โ‚›F

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (ฯ† : ๐“•.CrAnFieldOp) (ฯ†s ฯ†s' : List ๐“•.CrAnFieldOp) :    [ofCrAnListF ฯ†s, ofCrAnListF (ฯ† :: ฯ†s')]โ‚›F =    [ofCrAnListF ฯ†s, ofCrAnOpF ฯ†]โ‚›F * ofCrAnListF ฯ†s' +    ๐“ข(๐“• |>โ‚› ฯ†s, ๐“• |>โ‚› ฯ†)    โ€ข ofCrAnOpF ฯ† * [ofCrAnListF ฯ†s, ofCrAnListF ฯ†s']โ‚›F := by  rw [superCommuteF_ofCrAnListF_ofCrAnListF]  conv_rhs =>    lhs    rw [โ† ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF, sub_mul,      โ† ofCrAnListF_append]    rhs    rw [FieldStatistic.ofList_singleton, ofCrAnListF_append, ofCrAnListF_singleton, smul_mul_assoc,      mul_assoc, โ† ofCrAnListF_append]  conv_rhs =>    rhs    rw [superCommuteF_ofCrAnListF_ofCrAnListF, mul_sub, smul_mul_assoc]  simp only [List.cons_append, List.append_assoc, List.nil_append,    Algebra.mul_smul_comm, Algebra.smul_mul_assoc, sub_add_sub_cancel, sub_right_inj]  rw [โ† ofCrAnListF_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]  simp only [map_mul, mul_comm]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean:340-359

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Person-level attribution pending.

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