107 lines
3.6 KiB
Text
107 lines
3.6 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.SpaceTime.SL2C.Basic
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import HepLean.SpaceTime.LorentzVector.Complex.Modules
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import HepLean.Meta.Informal
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import Mathlib.RepresentationTheory.Rep
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import HepLean.SpaceTime.LorentzVector.Real.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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open SpaceTime
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namespace Lorentz
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open minkowskiMetric
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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 `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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/-!
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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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end Lorentz
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end
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