Merge pull request #37 from HEPLean/SpaceTime/LorentzGroup

feat: Add properties of lorentz algebra

HepLean.lean

@@ -58,6 +58,7 @@ import HepLean.GroupTheory.SO3.Basic
 import HepLean.SpaceTime.Basic
 import HepLean.SpaceTime.CliffordAlgebra
 import HepLean.SpaceTime.FourVelocity
+import HepLean.SpaceTime.LorentzAlgebra.Basic
 import HepLean.SpaceTime.LorentzGroup.Basic
 import HepLean.SpaceTime.LorentzGroup.Boosts
 import HepLean.SpaceTime.LorentzGroup.Orthochronous

HepLean/SpaceTime/LorentzAlgebra/Basic.lean

@@ -0,0 +1,115 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.Basic
+import HepLean.SpaceTime.Metric
+import Mathlib.Analysis.InnerProductSpace.Adjoint
+import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
+import Mathlib.Algebra.Lie.Classical
+import Mathlib.Algebra.Lie.TensorProduct
+import Mathlib.Analysis.InnerProductSpace.Adjoint
+/-!
+# The Lorentz Algebra
+
+
+-/
+
+
+namespace spaceTime
+open Matrix
+open TensorProduct
+
+/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 4) (Fin 4) ℝ`.  -/
+def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 4) (Fin 4) ℝ) :=
+  LieSubalgebra.map (Matrix.reindexLieEquiv (@finSumFinEquiv 1 3)).toLieHom
+  (LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
+
+
+namespace lorentzAlgebra
+
+lemma transpose_eta (A : lorentzAlgebra) :  A.1ᵀ * η  = - η * A.1  := by
+  have h := A.2
+  simp [lorentzAlgebra] at h
+  obtain ⟨B, hB1, hB2⟩ := h
+  simp [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] at hB1
+  apply (Equiv.apply_eq_iff_eq
+    (Matrix.reindexAlgEquiv ℝ (@finSumFinEquiv 1 3).symm).toEquiv).mp
+  erw [Matrix.reindexAlgEquiv_mul]
+  simp only [Nat.reduceAdd, reindexAlgEquiv_apply, Equiv.symm_symm, AlgEquiv.toEquiv_eq_coe,
+    EquivLike.coe_coe, map_neg, _root_.map_mul]
+  rw [← Matrix.transpose_reindex]
+  have h1 : (reindex finSumFinEquiv.symm finSumFinEquiv.symm) A = B :=
+    (Equiv.apply_eq_iff_eq_symm_apply (reindex finSumFinEquiv.symm finSumFinEquiv.symm)).mpr
+    (id hB2.symm)
+  rw [h1]
+  erw [η_reindex]
+  simpa using hB1
+
+lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 4) (Fin 4) ℝ}
+    (h :  Aᵀ * η  = - η * A) : A ∈ lorentzAlgebra := by
+  simp [lorentzAlgebra]
+  use (Matrix.reindexLieEquiv (@finSumFinEquiv 1 3)).symm A
+  apply And.intro
+  swap
+  change (reindexLieEquiv finSumFinEquiv) _ = _
+  simp only [Nat.reduceAdd, reindexLieEquiv_symm, reindexLieEquiv_apply, reindex_apply,
+    Equiv.symm_symm, submatrix_submatrix, Equiv.self_comp_symm, submatrix_id_id]
+  simp only [Nat.reduceAdd, reindexLieEquiv_symm, reindexLieEquiv_apply,
+    LieAlgebra.Orthogonal.so', mem_skewAdjointMatricesLieSubalgebra,
+    mem_skewAdjointMatricesSubmodule, IsSkewAdjoint, IsAdjointPair, mul_neg]
+  have h1 :=  (Equiv.apply_eq_iff_eq
+    (Matrix.reindexAlgEquiv ℝ (@finSumFinEquiv 1 3).symm).toEquiv).mpr h
+  erw [Matrix.reindexAlgEquiv_mul] at h1
+  simp only [Nat.reduceAdd, reindexAlgEquiv_apply, Equiv.symm_symm, AlgEquiv.toEquiv_eq_coe,
+    EquivLike.coe_coe, map_neg, _root_.map_mul] at h1
+  erw [η_reindex] at h1
+  simpa using h1
+
+
+lemma mem_iff {A : Matrix (Fin 4) (Fin 4) ℝ} : A ∈ lorentzAlgebra ↔
+    Aᵀ * η  = - η * A := by
+  apply Iff.intro
+  · intro h
+    exact transpose_eta ⟨A, h⟩
+  · intro h
+    exact mem_of_transpose_eta_eq_eta_mul_self h
+
+lemma mem_iff'  (A : Matrix (Fin 4) (Fin 4) ℝ) : A ∈ lorentzAlgebra ↔ A  = - η * Aᵀ * η := by
+  apply Iff.intro
+  intro h
+  rw [mul_assoc, mem_iff.mp h]
+  simp only [neg_mul, mul_neg, ← mul_assoc, η_sq, one_mul, neg_neg]
+  intro h
+  rw [mem_iff]
+  nth_rewrite 2 [h]
+  simp [← mul_assoc, η_sq]
+
+
+end lorentzAlgebra
+
+@[simps!]
+instance spaceTimeAsLieRingModule : LieRingModule lorentzAlgebra spaceTime where
+  bracket Λ x :=  Λ.1.mulVec  x
+  add_lie Λ1 Λ2 x := by
+    simp [add_mulVec]
+  lie_add Λ x1 x2 := by
+    simp only
+    exact mulVec_add _ _ _
+  leibniz_lie Λ1 Λ2 x := by
+    simp [mulVec_add, Bracket.bracket, sub_mulVec]
+
+@[simps!]
+instance spaceTimeAsLieModule : LieModule ℝ lorentzAlgebra spaceTime where
+  smul_lie r Λ x  := by
+    simp [Bracket.bracket, smul_mulVec_assoc]
+  lie_smul r Λ x := by
+    simp [Bracket.bracket]
+    rw [mulVec_smul]
+
+
+
+
+
+end spaceTime

HepLean/SpaceTime/Metric.lean

@@ -7,6 +7,7 @@ import HepLean.SpaceTime.Basic
 import Mathlib.Analysis.InnerProductSpace.Adjoint
 import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
 import Mathlib.Algebra.Lie.Classical
+import Mathlib.Algebra.Lie.TensorProduct
 /-!
 # Spacetime Metric
 
@@ -22,6 +23,7 @@ open Manifold
 open Matrix
 open Complex
 open ComplexConjugate
+open TensorProduct
 
 /-- The metric as a `4×4` real matrix. -/
 def η : Matrix (Fin 4) (Fin 4) ℝ := Matrix.reindex finSumFinEquiv finSumFinEquiv
@@ -38,6 +40,10 @@ lemma η_block : η = Matrix.reindex finSumFinEquiv finSumFinEquiv (
   funext i j
   fin_cases i <;> fin_cases j <;> simp
 
+lemma η_reindex : (Matrix.reindex finSumFinEquiv finSumFinEquiv).symm η =
+    LieAlgebra.Orthogonal.indefiniteDiagonal (Fin 1) (Fin 3) ℝ :=
+  (Equiv.symm_apply_eq (reindex finSumFinEquiv finSumFinEquiv)).mpr rfl
+
 lemma η_explicit : η = !![(1 : ℝ), 0, 0, 0; 0, -1, 0, 0; 0, 0, -1, 0; 0, 0, 0, -1] := by
   rw [η_block]
   apply Matrix.ext
@@ -243,6 +249,8 @@ lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
 def quadraticForm : QuadraticForm ℝ spaceTime := ηLin.toQuadraticForm
 
 
+
+
 end spaceTime
 
 end