138 lines
4.8 KiB
Text
138 lines
4.8 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import HepLean.Lorentz.Group.Basic
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import HepLean.Meta.Informal.Basic
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import Mathlib.RepresentationTheory.Rep
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import HepLean.Lorentz.RealVector.Modules
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/-!
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# Real Lorentz vectors
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We define real Lorentz vectors in as representations of the Lorentz group.
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-/
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noncomputable section
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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namespace Lorentz
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open minkowskiMatrix
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/-- The representation of `LorentzGroup d` on real vectors corresponding to contravariant
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Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
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def Contr (d : ℕ) : Rep ℝ (LorentzGroup d) := Rep.of ContrMod.rep
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/-- The representation of contrvariant Lorentz vectors forms a topological space, induced
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by its equivalence to `Fin 1 ⊕ Fin d → ℝ`. -/
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instance : TopologicalSpace (Contr d) := TopologicalSpace.induced
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ContrMod.toFin1dℝEquiv (Pi.topologicalSpace)
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lemma continuous_contr {T : Type} [TopologicalSpace T] (f : T → Contr d)
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(h : Continuous (fun i => (f i).toFin1dℝ)) : Continuous f := by
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exact continuous_induced_rng.mpr h
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lemma contr_continuous {T : Type} [TopologicalSpace T] (f : Contr d → T)
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(h : Continuous (f ∘ (@ContrMod.toFin1dℝEquiv d).symm)) : Continuous f := by
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let x := Equiv.toHomeomorphOfIsInducing (@ContrMod.toFin1dℝEquiv d).toEquiv
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ContrMod.toFin1dℝEquiv_isInducing
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rw [← Homeomorph.comp_continuous_iff' x.symm]
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exact h
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/-- The representation of `LorentzGroup d` on real vectors corresponding to covariant
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Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
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def Co (d : ℕ) : Rep ℝ (LorentzGroup d) := Rep.of CoMod.rep
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open CategoryTheory.MonoidalCategory
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/-!
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## Isomorphism between contravariant and covariant Lorentz vectors
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-/
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/-- The morphism of representations from `Contr d` to `Co d` defined by multiplication
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with the metric. -/
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def Contr.toCo (d : ℕ) : Contr d ⟶ Co d where
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hom := {
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toFun := fun ψ => CoMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ),
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map_add' := by
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intro ψ ψ'
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simp only [map_add, mulVec_add]
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map_smul' := by
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intro r ψ
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simp only [_root_.map_smul, mulVec_smul, RingHom.id_apply]}
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comm g := by
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ext ψ : 2
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simp only [ModuleCat.coe_comp, Function.comp_apply]
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conv_lhs =>
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change CoMod.toFin1dℝEquiv.symm (η *ᵥ (g.1 *ᵥ ψ.toFin1dℝ))
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rw [mulVec_mulVec, LorentzGroup.minkowskiMatrix_comm, ← mulVec_mulVec]
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rfl
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/-- The morphism of representations from `Co d` to `Contr d` defined by multiplication
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with the metric. -/
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def Co.toContr (d : ℕ) : Co d ⟶ Contr d where
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hom := {
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toFun := fun ψ => ContrMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ),
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map_add' := by
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intro ψ ψ'
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simp only [map_add, mulVec_add]
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map_smul' := by
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intro r ψ
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simp only [_root_.map_smul, mulVec_smul, RingHom.id_apply]}
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comm g := by
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ext ψ : 2
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simp only [ModuleCat.coe_comp, Function.comp_apply]
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conv_lhs =>
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change ContrMod.toFin1dℝEquiv.symm (η *ᵥ ((LorentzGroup.transpose g⁻¹).1 *ᵥ ψ.toFin1dℝ))
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rw [mulVec_mulVec, ← LorentzGroup.comm_minkowskiMatrix, ← mulVec_mulVec]
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rfl
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/-- The isomorphism between `Contr d` and `Co d` induced by multiplication with the
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Minkowski metric. -/
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def contrIsoCo (d : ℕ) : Contr d ≅ Co d where
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hom := Contr.toCo d
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inv := Co.toContr d
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hom_inv_id := by
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ext ψ
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simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
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ModuleCat.id_apply]
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conv_lhs => change ContrMod.toFin1dℝEquiv.symm (η *ᵥ
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CoMod.toFin1dℝEquiv (CoMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ)))
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rw [LinearEquiv.apply_symm_apply, mulVec_mulVec, minkowskiMatrix.sq]
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simp
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inv_hom_id := by
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ext ψ
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simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
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ModuleCat.id_apply]
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conv_lhs => change CoMod.toFin1dℝEquiv.symm (η *ᵥ
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ContrMod.toFin1dℝEquiv (ContrMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ)))
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rw [LinearEquiv.apply_symm_apply, mulVec_mulVec, minkowskiMatrix.sq]
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simp
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/-!
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## Other properties
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-/
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namespace Contr
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open Lorentz
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lemma ρ_stdBasis (μ : Fin 1 ⊕ Fin 3) (Λ : LorentzGroup 3) :
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(Contr 3).ρ Λ (ContrMod.stdBasis μ) = ∑ j, Λ.1 j μ • ContrMod.stdBasis j := by
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change Λ *ᵥ ContrMod.stdBasis μ = ∑ j, Λ.1 j μ • ContrMod.stdBasis j
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apply ContrMod.ext
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simp only [toLinAlgEquiv_self, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
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Fin.isValue, Finset.sum_singleton, ContrMod.val_add, ContrMod.val_smul]
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end Contr
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end Lorentz
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end
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