refactor: Adjust Minkowski metric
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7 changed files with 232 additions and 89 deletions
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@ -3,8 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.MinkowskiMetric
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import HepLean.SpaceTime.LorentzVector.NormOne
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import HepLean.SpaceTime.MinkowskiMatrix
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/-!
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# The Lorentz Group
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@ -32,17 +31,16 @@ These matrices form the Lorentz group, which we will define in the next section
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-/
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variable {d : ℕ}
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open minkowskiMetric in
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/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/
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open minkowskiMatrix in
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/-- The Lorentz group is the subset of matrices for which
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`Λ * dual Λ = 1`. -/
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def LorentzGroup (d : ℕ) : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :=
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{Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ |
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∀ (x y : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ y⟫ₘ = ⟪x, y⟫ₘ}
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{Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ | Λ * dual Λ = 1}
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namespace LorentzGroup
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/-- Notation for the Lorentz group. -/
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scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
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@ -52,32 +50,11 @@ variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
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-/
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lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔
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∀ (x : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ x⟫ₘ = ⟪x, x⟫ₘ := by
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refine Iff.intro (fun h x => h x x) (fun h x y => ?_)
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have hp := h (x + y)
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have hn := h (x - y)
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rw [mulVec_add] at hp
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rw [mulVec_sub] at hn
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simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn
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rw [symm (Λ *ᵥ y) (Λ *ᵥ x), symm y x] at hp hn
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linear_combination hp / 4 + -1 * hn / 4
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lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔
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∀ (x y : LorentzVector d), ⟪x, (dual Λ * Λ) *ᵥ y⟫ₘ = ⟪x, y⟫ₘ := by
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refine Iff.intro (fun h x y ↦ ?_) (fun h x y ↦ ?_)
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· have h1 := h x y
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rw [← dual_mulVec_right, mulVec_mulVec] at h1
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exact h1
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· rw [← dual_mulVec_right, mulVec_mulVec]
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exact h x y
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lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
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rfl
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lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
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rw [mem_iff_on_right, matrix_eq_id_iff]
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exact forall_comm
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lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
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rw [mem_iff_dual_mul_self]
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rw [mem_iff_self_mul_dual]
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exact mul_eq_one_comm
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lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d := by
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@ -108,6 +85,29 @@ lemma dual_mem (h : Λ ∈ LorentzGroup d) : dual Λ ∈ LorentzGroup d := by
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rw [mem_iff_dual_mul_self, dual_dual]
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exact mem_iff_self_mul_dual.mp h
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/-
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lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔
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∀ (x : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ x⟫ₘ = ⟪x, x⟫ₘ := by
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refine Iff.intro (fun h x => h x x) (fun h x y => ?_)
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have hp := h (x + y)
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have hn := h (x - y)
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rw [mulVec_add] at hp
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rw [mulVec_sub] at hn
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simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn
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rw [symm (Λ *ᵥ y) (Λ *ᵥ x), symm y x] at hp hn
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linear_combination hp / 4 + -1 * hn / 4
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lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔
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∀ (x y : LorentzVector d), ⟪x, (dual Λ * Λ) *ᵥ y⟫ₘ = ⟪x, y⟫ₘ := by
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refine Iff.intro (fun h x y ↦ ?_) (fun h x y ↦ ?_)
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· have h1 := h x y
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rw [← dual_mulVec_right, mulVec_mulVec] at h1
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exact h1
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· rw [← dual_mulVec_right, mulVec_mulVec]
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exact h x y
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-/
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end LorentzGroup
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/-!
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@ -131,7 +131,6 @@ instance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype
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namespace LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {Λ Λ' : LorentzGroup d}
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@ -285,43 +284,6 @@ lemma toGL_embedding : IsEmbedding (@toGL d).toFun where
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instance : TopologicalGroup (LorentzGroup d) :=
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IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
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section
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open LorentzVector
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/-!
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# To a norm one Lorentz vector
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-/
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/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/
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@[simps!]
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def toNormOneLorentzVector (Λ : LorentzGroup d) : NormOneLorentzVector d :=
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⟨Λ.1 *ᵥ timeVec, by rw [NormOneLorentzVector.mem_iff, Λ.2, minkowskiMetric.on_timeVec]⟩
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/-!
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# The time like element
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-/
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/-- The time like element of a Lorentz matrix. -/
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@[simp]
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def timeComp (Λ : LorentzGroup d) : ℝ := Λ.1 (Sum.inl 0) (Sum.inl 0)
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lemma timeComp_eq_toNormOneLorentzVector : timeComp Λ = (toNormOneLorentzVector Λ).1.time := by
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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rfl
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lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
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⟪toNormOneLorentzVector (transpose Λ), (toNormOneLorentzVector Λ').1.spaceReflection⟫ₘ := by
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simp only [timeComp, Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fintype.sum_sum_type,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton, toNormOneLorentzVector,
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transpose, timeVec, right_spaceReflection, time, space, PiLp.inner_apply, Function.comp_apply,
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RCLike.inner_apply, conj_trivial]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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simp
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/-!
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## To Complex matrices
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@ -387,6 +349,46 @@ lemma toComplex_mulVec_ofReal (v : Fin 1 ⊕ Fin d → ℝ) (Λ : LorentzGroup d
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funext i
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rw [← RingHom.map_mulVec]
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rfl
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/-!
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/-!
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# To a norm one Lorentz vector
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-/
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/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/
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@[simps!]
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def toNormOneLorentzVector (Λ : LorentzGroup d) : NormOneLorentzVector d :=
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⟨Λ.1 *ᵥ timeVec, by rw [NormOneLorentzVector.mem_iff, Λ.2, minkowskiMetric.on_timeVec]⟩
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/-!
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# The time like element
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-/
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/-- The time like element of a Lorentz matrix. -/
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@[simp]
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def timeComp (Λ : LorentzGroup d) : ℝ := Λ.1 (Sum.inl 0) (Sum.inl 0)
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lemma timeComp_eq_toNormOneLorentzVector : timeComp Λ = (toNormOneLorentzVector Λ).1.time := by
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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rfl
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lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
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⟪toNormOneLorentzVector (transpose Λ), (toNormOneLorentzVector Λ').1.spaceReflection⟫ₘ := by
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simp only [timeComp, Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fintype.sum_sum_type,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton, toNormOneLorentzVector,
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transpose, timeVec, right_spaceReflection, time, space, PiLp.inner_apply, Function.comp_apply,
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RCLike.inner_apply, conj_trivial]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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simp
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-/
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/-- The parity transformation. -/
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def parity : LorentzGroup d := ⟨minkowskiMatrix, by
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rw [mem_iff_dual_mul_self]
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simp only [dual_eta, minkowskiMatrix.sq]⟩
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end
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end LorentzGroup
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@ -21,7 +21,6 @@ open ComplexConjugate
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namespace LorentzGroup
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open minkowskiMetric
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open minkowskiMatrix
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variable {d : ℕ}
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@ -43,8 +43,8 @@ lemma SO3ToMatrix_injective : Function.Injective SO3ToMatrix := by
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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' := Subtype.eq <|
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(minkowskiMetric.matrix_eq_iff_eq_forall _ ↑1).mpr fun w => congrFun rfl
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map_one' := Subtype.eq <| by
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simp only [SO3ToMatrix, SO3Group_one_coe, Matrix.fromBlocks_one, lorentzGroupIsGroup_one_coe]
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map_mul' A B := Subtype.eq <| by
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simp only [SO3ToMatrix, SO3Group_mul_coe, lorentzGroupIsGroup_mul_coe,
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Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mul,
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