153 lines
5.7 KiB
Text
153 lines
5.7 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.Modules
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import HepLean.Meta.Informal
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import Mathlib.RepresentationTheory.Rep
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import HepLean.SpaceTime.PauliMatrices.SelfAdjoint
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/-!
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# Complex Lorentz vectors
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We define complex Lorentz vectors in 4d space-time as representations of SL(2, C).
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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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/-- The representation of `SL(2, ℂ)` on complex vectors corresponding to contravariant
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Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
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def complexContr : Rep ℂ SL(2, ℂ) := Rep.of ContrℂModule.SL2CRep
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/-- The representation of `SL(2, ℂ)` on complex vectors corresponding to contravariant
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Lorentz vectors. In index notation these have a down index `ψⁱ`. -/
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def complexCo : Rep ℂ SL(2, ℂ) := Rep.of CoℂModule.SL2CRep
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/-- The standard basis of complex contravariant Lorentz vectors. -/
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def complexContrBasis : Basis (Fin 1 ⊕ Fin 3) ℂ complexContr := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ ContrℂModule.toFin13ℂFun)
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@[simp]
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lemma complexContrBasis_toFin13ℂ (i :Fin 1 ⊕ Fin 3) :
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(complexContrBasis i).toFin13ℂ = Pi.single i 1 := by
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simp only [complexContrBasis, Basis.coe_ofEquivFun]
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rw [Lorentz.ContrℂModule.toFin13ℂ]
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rfl
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@[simp]
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lemma complexContrBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
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(LinearMap.toMatrix complexContrBasis complexContrBasis) (complexContr.ρ M) i j =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M)) i j := by
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rw [LinearMap.toMatrix_apply]
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simp only [complexContrBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
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change (((LorentzGroup.toComplex (SL2C.toLorentzGroup M))) *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, transpose_apply, mul_one]
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lemma complexContrBasis_ρ_val (M : SL(2,ℂ)) (v : complexContr) :
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((complexContr.ρ M) v).val =
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LorentzGroup.toComplex (SL2C.toLorentzGroup M) *ᵥ v.val := by
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rfl
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/-- The standard basis of complex contravariant Lorentz vectors indexed by `Fin 4`. -/
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def complexContrBasisFin4 : Basis (Fin 4) ℂ complexContr :=
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Basis.reindex complexContrBasis finSumFinEquiv
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/-- The standard basis of complex covariant Lorentz vectors. -/
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def complexCoBasis : Basis (Fin 1 ⊕ Fin 3) ℂ complexCo := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ CoℂModule.toFin13ℂFun)
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@[simp]
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lemma complexCoBasis_toFin13ℂ (i :Fin 1 ⊕ Fin 3) : (complexCoBasis i).toFin13ℂ = Pi.single i 1 := by
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simp only [complexCoBasis, Basis.coe_ofEquivFun]
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rw [Lorentz.CoℂModule.toFin13ℂ]
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rfl
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@[simp]
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lemma complexCoBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
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(LinearMap.toMatrix complexCoBasis complexCoBasis) (complexCo.ρ M) i j =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ i j := by
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rw [LinearMap.toMatrix_apply]
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simp only [complexCoBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
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change ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, transpose_apply, mul_one]
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/-- The standard basis of complex covariant Lorentz vectors indexed by `Fin 4`. -/
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def complexCoBasisFin4 : Basis (Fin 4) ℂ complexCo :=
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Basis.reindex complexCoBasis finSumFinEquiv
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/-!
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## Relation to real
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-/
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/-- The semilinear map including real Lorentz vectors into complex contravariant
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lorentz vectors. -/
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def inclCongrRealLorentz : LorentzVector 3 →ₛₗ[Complex.ofReal] complexContr where
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toFun v := {val := ofReal ∘ v}
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map_add' x y := by
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apply Lorentz.ContrℂModule.ext
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rw [Lorentz.ContrℂModule.val_add]
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funext i
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simp only [Function.comp_apply, ofReal_eq_coe, Pi.add_apply]
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change ofReal (x i + y i) = _
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simp only [ofReal_eq_coe, ofReal_add]
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map_smul' c x := by
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apply Lorentz.ContrℂModule.ext
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rw [Lorentz.ContrℂModule.val_smul]
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funext i
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simp only [Function.comp_apply, ofReal_eq_coe, Pi.smul_apply]
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change ofReal (c • x i) = _
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simp only [smul_eq_mul, ofReal_eq_coe, ofReal_mul]
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lemma inclCongrRealLorentz_val (v : LorentzVector 3) :
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(inclCongrRealLorentz v).val = ofReal ∘ v := rfl
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lemma complexContrBasis_of_real (i : Fin 1 ⊕ Fin 3) :
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(complexContrBasis i) = inclCongrRealLorentz (LorentzVector.stdBasis i) := by
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apply Lorentz.ContrℂModule.ext
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simp only [complexContrBasis, Basis.coe_ofEquivFun, inclCongrRealLorentz, LorentzVector.stdBasis,
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LinearMap.coe_mk, AddHom.coe_mk]
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ext j
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simp only [Function.comp_apply, ofReal_eq_coe]
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erw [Pi.basisFun_apply]
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change (Pi.single i 1) j = _
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exact Eq.symm (Pi.apply_single (fun _ => ofReal') (congrFun rfl) i 1 j)
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lemma inclCongrRealLorentz_ρ (M : SL(2, ℂ)) (v : LorentzVector 3) :
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(complexContr.ρ M) (inclCongrRealLorentz v) =
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inclCongrRealLorentz (SL2C.repLorentzVector M v) := by
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apply Lorentz.ContrℂModule.ext
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rw [complexContrBasis_ρ_val, inclCongrRealLorentz_val, inclCongrRealLorentz_val]
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rw [LorentzGroup.toComplex_mulVec_ofReal]
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apply congrArg
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simp only [SL2C.toLorentzGroup_apply_coe]
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rw [SL2C.repLorentzVector_apply_eq_mulVec]
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rfl
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lemma SL2CRep_ρ_basis (M : SL(2, ℂ)) (i : Fin 1 ⊕ Fin 3) :
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(complexContr.ρ M) (complexContrBasis i) =
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∑ j, (SL2C.toLorentzGroup M).1 j i •
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complexContrBasis j := by
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rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ, SL2C.repLorentzVector_stdBasis, map_sum]
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apply congrArg
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funext j
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simp only [LinearMap.map_smulₛₗ, ofReal_eq_coe, coe_smul]
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rw [complexContrBasis_of_real]
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end Lorentz
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end
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