PhysLean/HepLean/SpaceTime/LorentzGroup/Rotations.lean

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 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 to arbitrary dimensions.

-/
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 [minkowskiMetric.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' := by
    apply Subtype.eq
    simp only [SO3ToMatrix, Nat.reduceAdd, Matrix.reindex_apply, lorentzGroupIsGroup_one_coe]
    erw [Matrix.fromBlocks_one]
  map_mul' A B := by
    apply Subtype.eq
    simp [Matrix.fromBlocks_multiply]


end LorentzGroup


end