PhysLean/HepLean/Lorentz/RealVector/Basic.lean

/-
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.RealVector.Modules
/-!

# Real Lorentz vectors

We define real Lorentz vectors in as representations of the Lorentz group.

-/

noncomputable section

open Matrix
open MatrixGroups
open Complex
open TensorProduct

namespace Lorentz
open minkowskiMatrix
/-- The representation of `LorentzGroup d` on real vectors corresponding to contravariant
  Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
def Contr (d : ℕ) : Rep ℝ (LorentzGroup d) := Rep.of ContrMod.rep

/-- The representation of contrvariant Lorentz vectors forms a topological space, induced
  by its equivalence to `Fin 1 ⊕ Fin d → ℝ`. -/
instance : TopologicalSpace (Contr d) := TopologicalSpace.induced
  ContrMod.toFin1dℝEquiv (Pi.topologicalSpace)

lemma continuous_contr {T : Type} [TopologicalSpace T] (f : T → Contr d)
    (h : Continuous (fun i => (f i).toFin1dℝ)) : Continuous f := by
  exact continuous_induced_rng.mpr h

lemma contr_continuous {T : Type} [TopologicalSpace T] (f : Contr d → T)
    (h : Continuous (f ∘ (@ContrMod.toFin1dℝEquiv d).symm)) : Continuous f := by
  let x := Equiv.toHomeomorphOfIsInducing (@ContrMod.toFin1dℝEquiv d).toEquiv
    ContrMod.toFin1dℝEquiv_isInducing
  rw [← Homeomorph.comp_continuous_iff' x.symm]
  exact h

/-- The representation of `LorentzGroup d` on real vectors corresponding to covariant
  Lorentz vectors. In index notation these have an up index `ψⁱ`. -/
def Co (d : ℕ) : Rep ℝ (LorentzGroup d) := Rep.of CoMod.rep

open CategoryTheory.MonoidalCategory

/-!

## Isomorphism between contravariant and covariant Lorentz vectors

-/

/-- The morphism of representations from `Contr d` to `Co d` defined by multiplication
  with the metric. -/
def Contr.toCo (d : ℕ) : Contr d ⟶ Co d where
  hom := {
    toFun := fun ψ => CoMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ),
    map_add' := by
      intro ψ ψ'
      simp only [map_add, mulVec_add]
    map_smul' := by
      intro r ψ
      simp only [_root_.map_smul, mulVec_smul, RingHom.id_apply]}
  comm g := by
    ext ψ : 2
    simp only [ModuleCat.coe_comp, Function.comp_apply]
    conv_lhs =>
      change CoMod.toFin1dℝEquiv.symm (η *ᵥ (g.1 *ᵥ ψ.toFin1dℝ))
      rw [mulVec_mulVec, LorentzGroup.minkowskiMatrix_comm, ← mulVec_mulVec]
    rfl

/-- The morphism of representations from `Co d` to `Contr d` defined by multiplication
  with the metric. -/
def Co.toContr (d : ℕ) : Co d ⟶ Contr d where
  hom := {
    toFun := fun ψ => ContrMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ),
    map_add' := by
      intro ψ ψ'
      simp only [map_add, mulVec_add]
    map_smul' := by
      intro r ψ
      simp only [_root_.map_smul, mulVec_smul, RingHom.id_apply]}
  comm g := by
    ext ψ : 2
    simp only [ModuleCat.coe_comp, Function.comp_apply]
    conv_lhs =>
      change ContrMod.toFin1dℝEquiv.symm (η *ᵥ ((LorentzGroup.transpose g⁻¹).1 *ᵥ ψ.toFin1dℝ))
      rw [mulVec_mulVec, ← LorentzGroup.comm_minkowskiMatrix, ← mulVec_mulVec]
    rfl

/-- The isomorphism between `Contr d` and `Co d` induced by multiplication with the
  Minkowski metric. -/
def contrIsoCo (d : ℕ) : Contr d ≅ Co d where
  hom := Contr.toCo d
  inv := Co.toContr d
  hom_inv_id := by
    ext ψ
    simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
      ModuleCat.id_apply]
    conv_lhs => change ContrMod.toFin1dℝEquiv.symm (η *ᵥ
      CoMod.toFin1dℝEquiv (CoMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ)))
    rw [LinearEquiv.apply_symm_apply, mulVec_mulVec, minkowskiMatrix.sq]
    simp
  inv_hom_id := by
    ext ψ
    simp only [Action.comp_hom, ModuleCat.coe_comp, Function.comp_apply, Action.id_hom,
      ModuleCat.id_apply]
    conv_lhs => change CoMod.toFin1dℝEquiv.symm (η *ᵥ
      ContrMod.toFin1dℝEquiv (ContrMod.toFin1dℝEquiv.symm (η *ᵥ ψ.toFin1dℝ)))
    rw [LinearEquiv.apply_symm_apply, mulVec_mulVec, minkowskiMatrix.sq]
    simp

/-!

## Other properties

-/
namespace Contr

open Lorentz
lemma ρ_stdBasis (μ : Fin 1 ⊕ Fin 3) (Λ : LorentzGroup 3) :
    (Contr 3).ρ Λ (ContrMod.stdBasis μ) = ∑ j, Λ.1 j μ • ContrMod.stdBasis j := by
  change Λ *ᵥ ContrMod.stdBasis μ = ∑ j, Λ.1 j μ • ContrMod.stdBasis j
  apply ContrMod.ext
  simp only [toLinAlgEquiv_self, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
    Fin.isValue, Finset.sum_singleton, ContrMod.val_add, ContrMod.val_smul]

end Contr
end Lorentz
end