61 lines
1.9 KiB
Text
61 lines
1.9 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzGroup.Basic
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import HepLean.Mathematics.SO3.Basic
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import Mathlib.Topology.Constructions
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/-!
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# Rotations
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This file describes the embedding of `SO(3)` into `LorentzGroup 3`.
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## TODO
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Generalize to arbitrary dimensions.
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-/
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noncomputable section
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namespace LorentzGroup
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open GroupTheory
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/-- Given a element of `SO(3)` the matrix corresponding to a space-time rotation. -/
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@[simp]
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def SO3ToMatrix (A : SO(3)) : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ :=
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(Matrix.fromBlocks 1 0 0 A.1)
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lemma SO3ToMatrix_in_LorentzGroup (A : SO(3)) : SO3ToMatrix A ∈ LorentzGroup 3 := by
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rw [LorentzGroup.mem_iff_dual_mul_self]
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simp only [minkowskiMetric.dual, minkowskiMatrix.as_block, SO3ToMatrix,
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Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,
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Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul, Matrix.mul_one,
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neg_mul, one_mul, zero_add, Matrix.mul_neg, neg_zero, mul_neg, neg_neg,
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Matrix.mul_eq_one_comm.mpr A.2.2, Matrix.fromBlocks_one]
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lemma SO3ToMatrix_injective : Function.Injective SO3ToMatrix := by
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intro A B h
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erw [Equiv.apply_eq_iff_eq] at h
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have h1 := congrArg Matrix.toBlocks₂₂ h
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erw [Matrix.toBlocks_fromBlocks₂₂, Matrix.toBlocks_fromBlocks₂₂] at h1
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apply Subtype.eq
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exact h1
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/-- Given a element of `SO(3)` the element of the Lorentz group corresponding to a
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space-time rotation. -/
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def SO3ToLorentz : SO(3) →* LorentzGroup 3 where
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toFun A := ⟨SO3ToMatrix A, SO3ToMatrix_in_LorentzGroup A⟩
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map_one' := by
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apply Subtype.eq
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simp only [SO3ToMatrix, Nat.reduceAdd, Matrix.reindex_apply, lorentzGroupIsGroup_one_coe]
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erw [Matrix.fromBlocks_one]
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map_mul' A B := by
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apply Subtype.eq
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simp [Matrix.fromBlocks_multiply]
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end LorentzGroup
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end
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