2024-07-29 16:54:59 -04:00
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzVector.Basic
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import HepLean.SpaceTime.LorentzGroup.Basic
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import Mathlib.RepresentationTheory.Basic
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/-!
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# Lorentz group action on Lorentz vectors.
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-/
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noncomputable section
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namespace LorentzVector
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variable {d : ℕ} (v : LorentzVector d)
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/-- The contravariant action of the Lorentz group on a Lorentz vector. -/
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def rep : Representation ℝ (LorentzGroup d) (LorentzVector d) where
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toFun g := Matrix.toLinAlgEquiv e g
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2024-08-30 13:40:32 -04:00
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map_one' := (MulEquivClass.map_eq_one_iff (Matrix.toLinAlgEquiv e)).mpr rfl
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2024-07-29 16:54:59 -04:00
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map_mul' x y := by
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simp only [lorentzGroupIsGroup_mul_coe, map_mul]
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2024-07-30 07:51:07 -04:00
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open Matrix in
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lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v := rfl
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2024-07-31 08:52:09 -04:00
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lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
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rep g (stdBasis μ) = ∑ ν, g.1.transpose μ ν • stdBasis ν := by
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simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, decomp_stdBasis']
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funext ν
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2024-09-04 15:33:54 -04:00
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simp only [stdBasis, Matrix.transpose_apply]
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2024-09-04 06:28:46 -04:00
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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rfl
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2024-07-31 08:52:09 -04:00
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2024-07-29 16:54:59 -04:00
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end LorentzVector
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end
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