PhysLean/HepLean/Lorentz/ComplexVector/Modules.lean

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/-
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.Meta.Informal
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import HepLean.Lorentz.SL2C.Basic
import Mathlib.RepresentationTheory.Rep
import Mathlib.Logic.Equiv.TransferInstance
/-!
## Modules associated with complex Lorentz vectors
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We define the modules underlying complex Lorentz vectors.
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These definitions are preludes to the definitions of
`Lorentz.complexContr` and `Lorentz.complexCo`.
-/
namespace Lorentz
noncomputable section
open Matrix
open MatrixGroups
open Complex
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/-- The module for contravariant (up-index) complex Lorentz vectors. -/
structure ContrModule where
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/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → `. -/
val : Fin 1 ⊕ Fin 3 →
namespace ContrModule
/-- The equivalence between `ContrModule` and `Fin 1 ⊕ Fin 3 → `. -/
def toFin13Fun : ContrModule ≃ (Fin 1 ⊕ Fin 3 → ) where
toFun v := v.val
invFun f := ⟨f⟩
left_inv _ := rfl
right_inv _ := rfl
/-- The instance of `AddCommMonoid` on `ContrModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : AddCommMonoid ContrModule := Equiv.addCommMonoid toFin13Fun
/-- The instance of `AddCommGroup` on `ContrModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : AddCommGroup ContrModule := Equiv.addCommGroup toFin13Fun
/-- The instance of `Module` on `ContrModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : Module ContrModule := Equiv.module toFin13Fun
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@[ext]
lemma ext (ψ ψ' : ContrModule) (h : ψ.val = ψ'.val) : ψ = ψ' := by
cases ψ
cases ψ'
subst h
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rfl
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@[simp]
lemma val_add (ψ ψ' : ContrModule) : (ψ + ψ').val = ψ.val + ψ'.val := rfl
@[simp]
lemma val_smul (r : ) (ψ : ContrModule) : (r • ψ).val = r • ψ.val := rfl
/-- The linear equivalence between `ContrModule` and `(Fin 1 ⊕ Fin 3 → )`. -/
@[simps!]
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def toFin13Equiv : ContrModule ≃ₗ[] (Fin 1 ⊕ Fin 3 → ) :=
Equiv.linearEquiv toFin13Fun
/-- The underlying element of `Fin 1 ⊕ Fin 3 → ` of a element in `ContrModule` defined
through the linear equivalence `toFin13Equiv`. -/
abbrev toFin13 (ψ : ContrModule) := toFin13Equiv ψ
/-- The representation of the Lorentz group on `ContrModule`. -/
def lorentzGroupRep : Representation (LorentzGroup 3) ContrModule where
toFun M := {
toFun := fun v => toFin13Equiv.symm (LorentzGroup.toComplex M *ᵥ v.toFin13),
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' M N := by
ext i
simp
/-- The representation of the SL(2, ) on `ContrModule` induced by the representation of the
Lorentz group. -/
def SL2CRep : Representation SL(2, ) ContrModule :=
MonoidHom.comp lorentzGroupRep SpaceTime.SL2C.toLorentzGroup
end ContrModule
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/-- The module for covariant (up-index) complex Lorentz vectors. -/
structure CoModule where
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/-- The underlying value as a vector `Fin 1 ⊕ Fin 3 → `. -/
val : Fin 1 ⊕ Fin 3 →
namespace CoModule
/-- The equivalence between `CoModule` and `Fin 1 ⊕ Fin 3 → `. -/
def toFin13Fun : CoModule ≃ (Fin 1 ⊕ Fin 3 → ) where
toFun v := v.val
invFun f := ⟨f⟩
left_inv _ := rfl
right_inv _ := rfl
/-- The instance of `AddCommMonoid` on `CoModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : AddCommMonoid CoModule := Equiv.addCommMonoid toFin13Fun
/-- The instance of `AddCommGroup` on `CoModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : AddCommGroup CoModule := Equiv.addCommGroup toFin13Fun
/-- The instance of `Module` on `CoModule` defined via its equivalence
with `Fin 1 ⊕ Fin 3 → `. -/
instance : Module CoModule := Equiv.module toFin13Fun
/-- The linear equivalence between `CoModule` and `(Fin 1 ⊕ Fin 3 → )`. -/
@[simps!]
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def toFin13Equiv : CoModule ≃ₗ[] (Fin 1 ⊕ Fin 3 → ) :=
Equiv.linearEquiv toFin13Fun
/-- The underlying element of `Fin 1 ⊕ Fin 3 → ` of a element in `CoModule` defined
through the linear equivalence `toFin13Equiv`. -/
abbrev toFin13 (ψ : CoModule) := toFin13Equiv ψ
/-- The representation of the Lorentz group on `CoModule`. -/
def lorentzGroupRep : Representation (LorentzGroup 3) CoModule where
toFun M := {
toFun := fun v => toFin13Equiv.symm ((LorentzGroup.toComplex M)⁻¹ᵀ *ᵥ v.toFin13),
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' 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 _ ?_) _)
simp only [_root_.map_mul]
rw [Matrix.mul_inv_rev]
exact transpose_mul _ _
/-- The representation of the SL(2, ) on `ContrModule` induced by the representation of the
Lorentz group. -/
def SL2CRep : Representation SL(2, ) CoModule :=
MonoidHom.comp lorentzGroupRep SpaceTime.SL2C.toLorentzGroup
end CoModule
end
end Lorentz