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refactor: Move Lorentz Group

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jstoobysmith 2024-11-09 17:29:43 +00:00
parent 5cc188146f
commit 573ea890cd
13 changed files with 20 additions and 20 deletions
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55
HepLean/SpaceTime/LorentzGroup/Rotations.lean
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
import HepLean.Mathematics.SO3.Basic
import Mathlib.Topology.Constructions
/-!
# Rotations
This file describes the embedding of `SO(3)` into `LorentzGroup 3`.
-/
/-! TODO: Generalize the inclusion of rotations into LorentzGroup to abitary dimension. -/
noncomputable section
namespace LorentzGroup
open GroupTheory
/-- Given a element of `SO(3)` the matrix corresponding to a space-time rotation. -/
@[simp]
def SO3ToMatrix (A : SO(3)) : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ :=
(Matrix.fromBlocks 1 0 0 A.1)
lemma SO3ToMatrix_in_LorentzGroup (A : SO(3)) : SO3ToMatrix A ∈ LorentzGroup 3 := by
rw [LorentzGroup.mem_iff_dual_mul_self]
simp only [minkowskiMatrix.dual, minkowskiMatrix.as_block, SO3ToMatrix,
Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,
Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul, Matrix.mul_one,
neg_mul, one_mul, zero_add, Matrix.mul_neg, neg_zero, mul_neg, neg_neg,
Matrix.mul_eq_one_comm.mpr A.2.2, Matrix.fromBlocks_one]
lemma SO3ToMatrix_injective : Function.Injective SO3ToMatrix := by
intro A B h
erw [Equiv.apply_eq_iff_eq] at h
have h1 := congrArg Matrix.toBlocks₂₂ h
erw [Matrix.toBlocks_fromBlocks₂₂, Matrix.toBlocks_fromBlocks₂₂] at h1
apply Subtype.eq
exact h1
/-- Given a element of `SO(3)` the element of the Lorentz group corresponding to a
space-time rotation. -/
def SO3ToLorentz : SO(3) →* LorentzGroup 3 where
toFun A := ⟨SO3ToMatrix A, SO3ToMatrix_in_LorentzGroup A⟩
map_one' := Subtype.eq <| by
simp only [SO3ToMatrix, SO3Group_one_coe, Matrix.fromBlocks_one, lorentzGroupIsGroup_one_coe]
map_mul' A B := Subtype.eq <| by
simp only [SO3ToMatrix, SO3Group_mul_coe, lorentzGroupIsGroup_mul_coe,
Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul,
Matrix.mul_one, zero_add]
end LorentzGroup
end