`BookProof.Bosonic.commutator_field_Jfield` (h : BosonicCCR J a) (hJsq : ∀ v, J (J v) = -v) (v : V) : a v * a (J v) - a (J v) * a v = algebraMap ℂ R (-(Complex.I * ((‖v‖ ^ 2 : ℝ) :

Proved

BookProof.Bosonic.commutator_field_Jfield

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBosonicCCR.

BookProof.Bosonic.commutator_field_Jfield (h : BosonicCCR J a) (hJsq : ∀ v, J (J v) = -v) (v : V) : a v * a (J v) - a (J v) * a v = algebraMap ℂ R (-(Complex.I * ((‖v‖ ^ 2 : ℝ) : ℂ)))

Formalization note: Lean 4 identifier BookProof.Bosonic.commutator_field_Jfield.

Preamble

-- Generated from ChapterBosonicCCR.lean — theorem BookProof.Bosonic.commutator_field_Jfield
import Mathlib
import Definitions.Def_ChapterBosonicCCR
open BookProof.Bosonic


open RealInnerProductSpace


variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable {R : Type*} [Ring R] [StarRing R] [Algebra ℂ R] [Algebra ℝ R]
  [IsScalarTower ℝ ℂ R] [StarModule ℂ R]

variable {J : V →ₗ[ℝ] V} {a : V →ₗ[ℝ] R}

Formal statement

theorem BookProof.Bosonic.commutator_field_Jfield (h : BosonicCCR J a) (hJsq : ∀ v, J (J v) = -v) (v : V) :
    a v * a (J v) - a (J v) * a v
      = algebraMap ℂ R (-(Complex.I * ((‖v‖ ^ 2 : ℝ) : ℂ))) := by sorry

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