`BookProof.SmGaugeConnection.covD_commutator` {k : Fin 3 → ℝ} {g : ℝ} {T : Fin d → Matrix (Fin N) (Fin N) ℂ} {f : Fin d → Fin d → Fin d → ℝ} (hf : ClosesWithStructureConstants T f)

Proved

BookProof.SmGaugeConnection.covD_commutator

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmGaugeConnection.

BookProof.SmGaugeConnection.covD_commutator {k : Fin 3 → ℝ} {g : ℝ} {T : Fin d → Matrix (Fin N) (Fin N) ℂ} {f : Fin d → Fin d → Fin d → ℝ} (hf : ClosesWithStructureConstants T f) (A : Fin d → Fin 3 → ℝ) (j l : Fin 3) : covD k g T A j * covD k g T A l - covD k g T A l * covD k g T A j = ∑ c : Fin d, (-Complex.I * ((g : ℂ) ^ 2) * ((∑ a : Fin d, ∑ b : Fin d, f a b c * A a j * A b l : ℝ) : ℂ)) • T c

Formalization note: Lean 4 identifier BookProof.SmGaugeConnection.covD_commutator.

Preamble

-- Generated from ChapterSmGaugeConnection.lean — theorem BookProof.SmGaugeConnection.covD_commutator
import Definitions.Def_ChapterCPTHamiltonian
import Definitions.Def_ChapterSmCarAlgebra
import Definitions.Def_ChapterSmDiracYukawa
import Definitions.Def_ChapterSmDiracSpinor
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterSmGaugeConnection
import Definitions.Def_ChapterYangMillsSU3
open BookProof.YangMillsSU3
open BookProof.SmGaugeConnection



open Matrix Kronecker
open BookProof.YangMillsSU3 BookProof.ChapterCPTHamiltonian BookProof.SmCar
open BookProof.SmDiracYukawa BookProof.SmDiracSpinor BookProof.FarisLavine

noncomputable section

variable {N d : ℕ}

Formal statement

theorem BookProof.SmGaugeConnection.covD_commutator {k : Fin 3 → ℝ} {g : ℝ} {T : Fin d → Matrix (Fin N) (Fin N) ℂ}
    {f : Fin d → Fin d → Fin d → ℝ} (hf : ClosesWithStructureConstants T f)
    (A : Fin d → Fin 3 → ℝ) (j l : Fin 3) :
    covD k g T A j * covD k g T A l - covD k g T A l * covD k g T A j
      = ∑ c : Fin d,
          (-Complex.I * ((g : ℂ) ^ 2)
            * ((∑ a : Fin d, ∑ b : Fin d, f a b c * A a j * A b l : ℝ) : ℂ)) • T c := by sorry

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