297 lines
11 KiB
Text
297 lines
11 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.Lorentz.Weyl.Modules
|
||
/-!
|
||
|
||
# Weyl fermions
|
||
|
||
A good reference for the material in this file is:
|
||
https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf
|
||
|
||
-/
|
||
|
||
namespace Fermion
|
||
noncomputable section
|
||
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
open TensorProduct
|
||
|
||
/-- The vector space ℂ^2 carrying the fundamental representation of SL(2,C).
|
||
In index notation corresponds to a Weyl fermion with indices ψ^a. -/
|
||
def leftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
|
||
toFun := fun M => {
|
||
toFun := fun (ψ : LeftHandedModule) =>
|
||
LeftHandedModule.toFin2ℂEquiv.symm (M.1 *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' := fun M N => by
|
||
simp only [SpecialLinearGroup.coe_mul]
|
||
ext1 x
|
||
simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply,
|
||
mulVec_mulVec]}
|
||
|
||
/-- The standard basis on left-handed Weyl fermions. -/
|
||
def leftBasis : Basis (Fin 2) ℂ leftHanded := Basis.ofEquivFun
|
||
(Equiv.linearEquiv ℂ LeftHandedModule.toFin2ℂFun)
|
||
|
||
@[simp]
|
||
lemma leftBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
|
||
(LinearMap.toMatrix leftBasis leftBasis) (leftHanded.ρ M) i j = M.1 i j := by
|
||
rw [LinearMap.toMatrix_apply]
|
||
simp only [leftBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply]
|
||
change (M.1 *ᵥ (Pi.single j 1)) i = _
|
||
simp only [mulVec_single, mul_one]
|
||
|
||
@[simp]
|
||
lemma leftBasis_toFin2ℂ (i : Fin 2) : (leftBasis i).toFin2ℂ = Pi.single i 1 := by
|
||
simp only [leftBasis, Basis.coe_ofEquivFun]
|
||
rfl
|
||
|
||
/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
|
||
M → (M⁻¹)ᵀ. In index notation corresponds to a Weyl fermion with indices ψ_a. -/
|
||
def altLeftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
|
||
toFun := fun M => {
|
||
toFun := fun (ψ : AltLeftHandedModule) =>
|
||
AltLeftHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹)ᵀ *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' := fun M N => by
|
||
ext1 x
|
||
simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
|
||
LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
|
||
refine (congrFun (congrArg _ ?_) _)
|
||
rw [Matrix.mul_inv_rev]
|
||
exact transpose_mul _ _}
|
||
|
||
/-- The standard basis on alt-left-handed Weyl fermions. -/
|
||
def altLeftBasis : Basis (Fin 2) ℂ altLeftHanded := Basis.ofEquivFun
|
||
(Equiv.linearEquiv ℂ AltLeftHandedModule.toFin2ℂFun)
|
||
|
||
@[simp]
|
||
lemma altLeftBasis_toFin2ℂ (i : Fin 2) : (altLeftBasis i).toFin2ℂ = Pi.single i 1 := by
|
||
simp only [altLeftBasis, Basis.coe_ofEquivFun]
|
||
rfl
|
||
|
||
@[simp]
|
||
lemma altLeftBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
|
||
(LinearMap.toMatrix altLeftBasis altLeftBasis) (altLeftHanded.ρ M) i j = (M.1⁻¹)ᵀ i j := by
|
||
rw [LinearMap.toMatrix_apply]
|
||
simp only [altLeftBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
|
||
change ((M.1⁻¹)ᵀ *ᵥ (Pi.single j 1)) i = _
|
||
simp only [mulVec_single, transpose_apply, mul_one]
|
||
|
||
/-- The vector space ℂ^2 carrying the conjugate representation of SL(2,C).
|
||
In index notation corresponds to a Weyl fermion with indices ψ^{dot a}. -/
|
||
def rightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
|
||
toFun := fun M => {
|
||
toFun := fun (ψ : RightHandedModule) =>
|
||
RightHandedModule.toFin2ℂEquiv.symm (M.1.map star *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' := fun M N => by
|
||
ext1 x
|
||
simp only [SpecialLinearGroup.coe_mul, RCLike.star_def, Matrix.map_mul, LinearMap.coe_mk,
|
||
AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply, mulVec_mulVec]}
|
||
|
||
/-- The standard basis on right-handed Weyl fermions. -/
|
||
def rightBasis : Basis (Fin 2) ℂ rightHanded := Basis.ofEquivFun
|
||
(Equiv.linearEquiv ℂ RightHandedModule.toFin2ℂFun)
|
||
|
||
@[simp]
|
||
lemma rightBasis_toFin2ℂ (i : Fin 2) : (rightBasis i).toFin2ℂ = Pi.single i 1 := by
|
||
simp only [rightBasis, Basis.coe_ofEquivFun]
|
||
rfl
|
||
|
||
@[simp]
|
||
lemma rightBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
|
||
(LinearMap.toMatrix rightBasis rightBasis) (rightHanded.ρ M) i j = (M.1.map star) i j := by
|
||
rw [LinearMap.toMatrix_apply]
|
||
simp only [rightBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
|
||
change (M.1.map star *ᵥ (Pi.single j 1)) i = _
|
||
simp only [mulVec_single, transpose_apply, mul_one]
|
||
|
||
/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
|
||
M → (M⁻¹)^†.
|
||
In index notation this corresponds to a Weyl fermion with index `ψ_{dot a}`. -/
|
||
def altRightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
|
||
toFun := fun M => {
|
||
toFun := fun (ψ : AltRightHandedModule) =>
|
||
AltRightHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹).conjTranspose *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp [mulVec_add]
|
||
map_smul' := by
|
||
intro r ψ
|
||
simp [mulVec_smul]}
|
||
map_one' := by
|
||
ext i
|
||
simp
|
||
map_mul' := fun M N => by
|
||
ext1 x
|
||
simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply,
|
||
LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
|
||
refine (congrFun (congrArg _ ?_) _)
|
||
rw [Matrix.mul_inv_rev]
|
||
exact conjTranspose_mul _ _}
|
||
|
||
/-- The standard basis on alt-right-handed Weyl fermions. -/
|
||
def altRightBasis : Basis (Fin 2) ℂ altRightHanded := Basis.ofEquivFun
|
||
(Equiv.linearEquiv ℂ AltRightHandedModule.toFin2ℂFun)
|
||
|
||
@[simp]
|
||
lemma altRightBasis_toFin2ℂ (i : Fin 2) : (altRightBasis i).toFin2ℂ = Pi.single i 1 := by
|
||
simp only [altRightBasis, Basis.coe_ofEquivFun]
|
||
rfl
|
||
|
||
@[simp]
|
||
lemma altRightBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
|
||
(LinearMap.toMatrix altRightBasis altRightBasis) (altRightHanded.ρ M) i j =
|
||
((M.1⁻¹).conjTranspose) i j := by
|
||
rw [LinearMap.toMatrix_apply]
|
||
simp only [altRightBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
|
||
change ((M.1⁻¹).conjTranspose *ᵥ (Pi.single j 1)) i = _
|
||
simp only [mulVec_single, transpose_apply, mul_one]
|
||
|
||
/-!
|
||
|
||
## Equivalences between Weyl fermion vector spaces.
|
||
|
||
-/
|
||
|
||
/-- The morphism between the representation `leftHanded` and the representation
|
||
`altLeftHanded` defined by multiplying an element of
|
||
`leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`. -/
|
||
def leftHandedToAlt : leftHanded ⟶ altLeftHanded where
|
||
hom := {
|
||
toFun := fun ψ => AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp only [mulVec_add, LinearEquiv.map_add]
|
||
map_smul' := by
|
||
intro a ψ
|
||
simp only [mulVec_smul, LinearEquiv.map_smul]
|
||
rfl}
|
||
comm := by
|
||
intro M
|
||
refine LinearMap.ext (fun ψ => ?_)
|
||
change AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ M.1 *ᵥ ψ.val) =
|
||
AltLeftHandedModule.toFin2ℂEquiv.symm ((M.1⁻¹)ᵀ *ᵥ !![0, 1; -1, 0] *ᵥ ψ.val)
|
||
apply congrArg
|
||
rw [mulVec_mulVec, mulVec_mulVec, Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
|
||
refine congrFun (congrArg _ ?_) _
|
||
rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
|
||
Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
|
||
simp
|
||
|
||
lemma leftHandedToAlt_hom_apply (ψ : leftHanded) :
|
||
leftHandedToAlt.hom ψ =
|
||
AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ) := rfl
|
||
|
||
/-- The morphism from `altLeftHanded` to
|
||
`leftHanded` defined by multiplying an element of
|
||
altLeftHandedWeyl by the matrix `εₐ₁ₐ₂ = !![0, -1; 1, 0]`. -/
|
||
def leftHandedAltTo : altLeftHanded ⟶ leftHanded where
|
||
hom := {
|
||
toFun := fun ψ =>
|
||
LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ),
|
||
map_add' := by
|
||
intro ψ ψ'
|
||
simp only [map_add]
|
||
rw [mulVec_add, LinearEquiv.map_add]
|
||
map_smul' := by
|
||
intro a ψ
|
||
simp only [LinearEquiv.map_smul]
|
||
rw [mulVec_smul, LinearEquiv.map_smul]
|
||
rfl}
|
||
comm := by
|
||
intro M
|
||
refine LinearMap.ext (fun ψ => ?_)
|
||
change LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ (M.1⁻¹)ᵀ *ᵥ ψ.val) =
|
||
LeftHandedModule.toFin2ℂEquiv.symm (M.1 *ᵥ !![0, -1; 1, 0] *ᵥ ψ.val)
|
||
rw [EquivLike.apply_eq_iff_eq, mulVec_mulVec, mulVec_mulVec, Lorentz.SL2C.inverse_coe,
|
||
eta_fin_two M.1]
|
||
refine congrFun (congrArg _ ?_) _
|
||
rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
|
||
Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
|
||
simp
|
||
|
||
lemma leftHandedAltTo_hom_apply (ψ : altLeftHanded) :
|
||
leftHandedAltTo.hom ψ =
|
||
LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ) := rfl
|
||
|
||
/-- The equivalence between the representation `leftHanded` and the representation
|
||
`altLeftHanded` defined by multiplying an element of
|
||
`leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`. -/
|
||
def leftHandedAltEquiv : leftHanded ≅ altLeftHanded where
|
||
hom := leftHandedToAlt
|
||
inv := leftHandedAltTo
|
||
hom_inv_id := by
|
||
ext ψ
|
||
simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
|
||
ModuleCat.id_apply]
|
||
rw [leftHandedAltTo_hom_apply, leftHandedToAlt_hom_apply]
|
||
rw [AltLeftHandedModule.toFin2ℂ, LinearEquiv.apply_symm_apply, mulVec_mulVec]
|
||
rw [show (!![0, -1; (1 : ℂ), 0] * !![0, 1; -1, 0]) = 1 by simpa using Eq.symm one_fin_two]
|
||
rw [one_mulVec]
|
||
rfl
|
||
inv_hom_id := by
|
||
ext ψ
|
||
simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
|
||
ModuleCat.id_apply]
|
||
rw [leftHandedAltTo_hom_apply, leftHandedToAlt_hom_apply, LeftHandedModule.toFin2ℂ,
|
||
LinearEquiv.apply_symm_apply, mulVec_mulVec]
|
||
rw [show (!![0, (1 : ℂ); -1, 0] * !![0, -1; 1, 0]) = 1 by simpa using Eq.symm one_fin_two]
|
||
rw [one_mulVec]
|
||
rfl
|
||
|
||
/-- `leftHandedAltEquiv` acting on an element `ψ : leftHanded` corresponds
|
||
to multiplying `ψ` by the matrix `!![0, 1; -1, 0]`. -/
|
||
lemma leftHandedAltEquiv_hom_hom_apply (ψ : leftHanded) :
|
||
leftHandedAltEquiv.hom.hom ψ =
|
||
AltLeftHandedModule.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ) := rfl
|
||
|
||
/-- The inverse of `leftHandedAltEquiv` acting on an element`ψ : altLeftHanded` corresponds
|
||
to multiplying `ψ` by the matrix `!![0, -1; 1, 0]`. -/
|
||
lemma leftHandedAltEquiv_inv_hom_apply (ψ : altLeftHanded) :
|
||
leftHandedAltEquiv.inv.hom ψ =
|
||
LeftHandedModule.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ) := rfl
|
||
|
||
informal_definition rightHandedWeylAltEquiv where
|
||
math :≈ "The linear equiv between rightHandedWeyl and altRightHandedWeyl given
|
||
by multiplying an element of rightHandedWeyl by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`"
|
||
deps :≈ [``rightHanded, ``altRightHanded]
|
||
|
||
informal_lemma rightHandedWeylAltEquiv_equivariant where
|
||
math :≈ "The linear equiv rightHandedWeylAltEquiv is equivariant with respect to the
|
||
action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
|
||
deps :≈ [``rightHandedWeylAltEquiv]
|
||
|
||
end
|
||
|
||
end Fermion
|