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feat: Add Pauli-matrices as tensor.

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jstoobysmith 2024-10-16 10:39:11 +00:00
parent a60ade65f0
commit 691b7e112e
13 changed files with 989 additions and 154 deletions
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16
HepLean/SpaceTime/WeylFermion/Unit.lean
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@ -24,11 +24,11 @@ open Complex
open TensorProduct
open CategoryTheory.MonoidalCategory
/-- The left-alt-left unit `δₐᵃ` as an element of `(leftHanded ⊗ altLeftHanded).V`. -/
/-- The left-alt-left unit `δᵃₐ` as an element of `(leftHanded ⊗ altLeftHanded).V`. -/
def leftAltLeftUnitVal : (leftHanded ⊗ altLeftHanded).V :=
leftAltLeftToMatrix.symm 1
/-- The left-alt-left unit `δₐᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded `,
/-- The left-alt-left unit `δᵃₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded `,
manifesting the invariance under the `SL(2,ℂ)` action. -/
def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded where
hom := {
@ -55,11 +55,11 @@ def leftAltLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ altLeftHanded
apply congrArg
simp
/-- The alt-left-left unit `δᵃₐ` as an element of `(altLeftHanded ⊗ leftHanded).V`. -/
/-- The alt-left-left unit `δₐᵃ` as an element of `(altLeftHanded ⊗ leftHanded).V`. -/
def altLeftLeftUnitVal : (altLeftHanded ⊗ leftHanded).V :=
altLeftLeftToMatrix.symm 1
/-- The alt-left-left unit `δᵃₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded `,
/-- The alt-left-left unit `δₐᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded `,
manifesting the invariance under the `SL(2,ℂ)` action. -/
def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded where
hom := {
@ -87,12 +87,12 @@ def altLeftLeftUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ leftHanded
simp only [mul_one, ← transpose_mul, SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq,
one_ne_zero, not_false_eq_true, mul_nonsing_inv, transpose_one]
/-- The right-alt-right unit `δ_{dot a}^{dot a}` as an element of
/-- The right-alt-right unit `δ^{dot a}_{dot a}` as an element of
`(rightHanded ⊗ altRightHanded).V`. -/
def rightAltRightUnitVal : (rightHanded ⊗ altRightHanded).V :=
rightAltRightToMatrix.symm 1
/-- The right-alt-right unit `δ_{dot a}^{dot a}` as a morphism
/-- The right-alt-right unit `δ^{dot a}_{dot a}` as a morphism
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHanded`, manifesting
the invariance under the `SL(2,ℂ)` action. -/
def rightAltRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHanded where
@ -126,12 +126,12 @@ def rightAltRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ altRightHa
rw [@conjTranspose_nonsing_inv]
simp
/-- The alt-right-right unit `δ^{dot a}_{dot a}` as an element of
/-- The alt-right-right unit `δ_{dot a}^{dot a}` as an element of
`(rightHanded ⊗ altRightHanded).V`. -/
def altRightRightUnitVal : (altRightHanded ⊗ rightHanded).V :=
altRightRightToMatrix.symm 1
/-- The alt-right-right unit `δ^{dot a}_{dot a}` as a morphism
/-- The alt-right-right unit `δ_{dot a}^{dot a}` as a morphism
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHanded`, manifesting
the invariance under the `SL(2,ℂ)` action. -/
def altRightRightUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ rightHanded where