2024-05-21 06:21:51 -04:00
|
|
|
|
/-
|
|
|
|
|
|
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|
|
|
|
|
Released under Apache 2.0 license.
|
|
|
|
|
|
Authors: Joseph Tooby-Smith
|
|
|
|
|
|
-/
|
2024-06-25 07:06:32 -04:00
|
|
|
|
import HepLean.SpaceTime.LorentzGroup.Basic
|
2024-05-21 06:21:51 -04:00
|
|
|
|
import HepLean.GroupTheory.SO3.Basic
|
|
|
|
|
|
import Mathlib.Topology.Constructions
|
|
|
|
|
|
/-!
|
|
|
|
|
|
# Rotations
|
|
|
|
|
|
|
|
|
|
|
|
-/
|
|
|
|
|
|
noncomputable section
|
2024-06-26 11:54:02 -04:00
|
|
|
|
namespace SpaceTime
|
2024-05-21 06:21:51 -04:00
|
|
|
|
|
|
|
|
|
|
namespace lorentzGroup
|
|
|
|
|
|
open GroupTheory
|
|
|
|
|
|
|
2024-05-22 13:34:53 -04:00
|
|
|
|
/-- Given a element of `SO(3)` the matrix corresponding to a space-time rotation. -/
|
2024-05-22 09:18:12 -04:00
|
|
|
|
@[simp]
|
|
|
|
|
|
def SO3ToMatrix (A : SO(3)) : Matrix (Fin 4) (Fin 4) ℝ :=
|
|
|
|
|
|
Matrix.reindex finSumFinEquiv finSumFinEquiv (Matrix.fromBlocks 1 0 0 A.1)
|
|
|
|
|
|
|
2024-05-22 13:34:53 -04:00
|
|
|
|
lemma SO3ToMatrix_PreservesηLin (A : SO(3)) : PreservesηLin $ SO3ToMatrix A := by
|
|
|
|
|
|
rw [PreservesηLin.iff_matrix]
|
2024-05-22 09:18:12 -04:00
|
|
|
|
simp only [η_block, Nat.reduceAdd, Matrix.reindex_apply, SO3ToMatrix, Matrix.transpose_submatrix,
|
|
|
|
|
|
Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,
|
|
|
|
|
|
Matrix.submatrix_mul_equiv, 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, Matrix.submatrix_one_equiv]
|
2024-05-21 06:21:51 -04:00
|
|
|
|
|
2024-05-22 13:34:53 -04:00
|
|
|
|
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
|
|
|
|
|
|
rw [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) →* 𝓛 where
|
|
|
|
|
|
toFun A := ⟨SO3ToMatrix A, SO3ToMatrix_PreservesηLin A⟩
|
|
|
|
|
|
map_one' := by
|
|
|
|
|
|
apply Subtype.eq
|
|
|
|
|
|
simp only [SO3ToMatrix, Nat.reduceAdd, Matrix.reindex_apply, lorentzGroupIsGroup_one_coe]
|
|
|
|
|
|
erw [Matrix.fromBlocks_one]
|
|
|
|
|
|
exact Matrix.submatrix_one_equiv finSumFinEquiv.symm
|
|
|
|
|
|
map_mul' A B := by
|
|
|
|
|
|
apply Subtype.eq
|
|
|
|
|
|
simp [Matrix.fromBlocks_multiply]
|
|
|
|
|
|
|
2024-05-21 06:21:51 -04:00
|
|
|
|
|
|
|
|
|
|
end lorentzGroup
|
|
|
|
|
|
|
|
|
|
|
|
|
2024-06-26 11:54:02 -04:00
|
|
|
|
end SpaceTime
|
2024-05-21 06:21:51 -04:00
|
|
|
|
end
|
