159 lines
4.9 KiB
Text
159 lines
4.9 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 HepLean.SpaceTime.WeylFermion.Basic
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import HepLean.SpaceTime.WeylFermion.Contraction
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import Mathlib.LinearAlgebra.TensorProduct.Matrix
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import HepLean.SpaceTime.WeylFermion.Two
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/-!
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# Units of Weyl fermions
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We define the units for Weyl fermions, often denoted `δ` in the literature.
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-/
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namespace Fermion
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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 CategoryTheory.MonoidalCategory
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/-- The left-alt-left unit `δₐᵃ` as an element of `(leftHanded ⊗ altLeftHanded).V`. -/
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def leftAltLeftUnitVal : (leftHanded ⊗ altLeftHanded).V :=
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leftAltLeftToMatrix.symm 1
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/-- The left-alt-left unit `δₐᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded `,
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manifesting the invariance under the `SL(2,ℂ)` action. -/
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def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded where
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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • leftAltLeftUnitVal,
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map_add' := fun x y => by
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simp only [add_smul]
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp
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let x' : ℂ := x
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change x' • leftAltLeftUnitVal =
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)) (x' • leftAltLeftUnitVal)
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simp
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apply congrArg
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simp [leftAltLeftUnitVal]
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erw [leftAltLeftToMatrix_ρ_symm]
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apply congrArg
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simp
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/-- The alt-left-left unit `δᵃₐ` as an element of `(altLeftHanded ⊗ leftHanded).V`. -/
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def altLeftLeftUnitVal : (altLeftHanded ⊗ leftHanded).V :=
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altLeftLeftToMatrix.symm 1
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/-- The alt-left-left unit `δᵃₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded `,
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manifesting the invariance under the `SL(2,ℂ)` action. -/
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def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded where
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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • altLeftLeftUnitVal,
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map_add' := fun x y => by
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simp only [add_smul]
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp
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let x' : ℂ := x
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change x' • altLeftLeftUnitVal =
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)) (x' • altLeftLeftUnitVal)
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simp
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apply congrArg
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simp [altLeftLeftUnitVal]
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erw [altLeftLeftToMatrix_ρ_symm]
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apply congrArg
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simp only [mul_one, ← transpose_mul, SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq,
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one_ne_zero, not_false_eq_true, mul_nonsing_inv, transpose_one]
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/-- The right-alt-right unit `δ_{dot a}^{dot a}` as an element of
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`(rightHanded ⊗ altRightHanded).V`. -/
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def rightAltRightUnitVal : (rightHanded ⊗ altRightHanded).V :=
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rightAltRightToMatrix.symm 1
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/-- The right-alt-right unit `δ_{dot a}^{dot a}` as a morphism
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`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHanded`, manifesting
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the invariance under the `SL(2,ℂ)` action. -/
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def rightAltRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHanded where
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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • rightAltRightUnitVal,
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map_add' := fun x y => by
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simp only [add_smul]
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp
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let x' : ℂ := x
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change x' • rightAltRightUnitVal =
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(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) (x' • rightAltRightUnitVal)
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simp
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apply congrArg
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simp [rightAltRightUnitVal]
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erw [rightAltRightToMatrix_ρ_symm]
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apply congrArg
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simp
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symm
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refine transpose_eq_one.mp ?h.h.h.a
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simp
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change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
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rw [@conjTranspose_nonsing_inv]
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simp
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/-- The alt-right-right unit `δ^{dot a}_{dot a}` as an element of
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`(rightHanded ⊗ altRightHanded).V`. -/
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def altRightRightUnitVal : (altRightHanded ⊗ rightHanded).V :=
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altRightRightToMatrix.symm 1
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/-- The alt-right-right unit `δ^{dot a}_{dot a}` as a morphism
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`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHanded`, manifesting
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the invariance under the `SL(2,ℂ)` action. -/
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def altRightRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHanded where
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hom := {
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toFun := fun a =>
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let a' : ℂ := a
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a' • altRightRightUnitVal,
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map_add' := fun x y => by
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simp only [add_smul]
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map_smul' := fun m x => by
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simp only [smul_smul]
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rfl}
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comm M := by
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ext x : 2
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simp
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let x' : ℂ := x
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change x' • altRightRightUnitVal =
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(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) (x' • altRightRightUnitVal)
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simp
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apply congrArg
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simp [altRightRightUnitVal]
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erw [altRightRightToMatrix_ρ_symm]
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apply congrArg
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simp
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symm
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change (M.1)⁻¹ᴴ * (M.1)ᴴ = 1
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rw [@conjTranspose_nonsing_inv]
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simp
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end
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end Fermion
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