refactor: Lorentz tensors
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jstoobysmith
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7 changed files with 492 additions and 359 deletions
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@ -71,6 +71,10 @@ import HepLean.SpaceTime.LorentzGroup.Proper
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import HepLean.SpaceTime.LorentzGroup.Restricted
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import HepLean.SpaceTime.LorentzGroup.Rotations
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import HepLean.SpaceTime.LorentzTensor.Basic
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import HepLean.SpaceTime.LorentzTensor.Contractions
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import HepLean.SpaceTime.LorentzTensor.Fin
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import HepLean.SpaceTime.LorentzTensor.LorentzTensorStruct
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import HepLean.SpaceTime.LorentzTensor.RisingLowering
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import HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix
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import HepLean.SpaceTime.LorentzVector.Basic
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import HepLean.SpaceTime.LorentzVector.NormOne
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@ -4,12 +4,23 @@ 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 Mathlib.LinearAlgebra.PiTensorProduct
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import Mathlib.RepresentationTheory.Basic
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import Mathlib.LinearAlgebra.DFinsupp
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/-!
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# Structure of Lorentz Tensors
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# Structure of Tensors
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In this file we set up the basic structures we will use to define Lorentz tensors.
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This file sets up the structure `TensorStructure` which contains
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data of types (or `colors`) of indices, the dual of colors, the associated module,
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contraction between color modules, and the unit of contraction.
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This structure is extended to `DualizeTensorStructure` which adds a metric to the tensor structure,
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allowing a vector to be taken to its dual vector by contraction with a specified metric.
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The definition of `DualizeTensorStructure` can be found in
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`HepLean.SpaceTime.LorentzTensor.RisingLowering`.
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The structure `DualizeTensorStructure` is further extended in
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`HepLean.SpaceTime.LorentzTensor.LorentzTensorStruct` to add a group action on the tensor space,
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under which contraction and rising and lowering etc, are invariant.
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## References
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@ -24,8 +35,8 @@ open TensorProduct
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variable {R : Type} [CommSemiring R]
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/-- An initial structure specifying a tensor system (e.g. a system in which you can
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define real Lorentz tensors). -/
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structure PreTensorStructure (R : Type) [CommSemiring R] where
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define real Lorentz tensors or Einstein notation convention). -/
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structure TensorStructure (R : Type) [CommSemiring R] where
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/-- The allowed colors of indices.
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For example for a real Lorentz tensor these are `{up, down}`. -/
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Color : Type
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@ -41,10 +52,20 @@ structure PreTensorStructure (R : Type) [CommSemiring R] where
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colorModule_module : ∀ μ, Module R (ColorModule μ)
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/-- The contraction of a vector with a vector with dual color. -/
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contrDual : ∀ μ, ColorModule μ ⊗[R] ColorModule (τ μ) →ₗ[R] R
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/-- The contraction is symmetric. -/
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contrDual_symm : ∀ μ x y, (contrDual μ) (x ⊗ₜ[R] y) =
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(contrDual (τ μ)) (y ⊗ₜ[R] (Equiv.cast (congrArg ColorModule (τ_involutive μ).symm) x))
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/-- The unit of the contraction. -/
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unit : (μ : Color) → ColorModule (τ μ) ⊗[R] ColorModule μ
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/-- The unit is a right identity. -/
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unit_rid : ∀ μ (x : ColorModule μ),
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TensorProduct.lid R _
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(TensorProduct.map (contrDual μ) (LinearEquiv.refl R (ColorModule μ)).toLinearMap
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((TensorProduct.assoc R _ _ _).symm (x ⊗ₜ[R] unit μ))) = x
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namespace PreTensorStructure
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namespace TensorStructure
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variable (𝓣 : PreTensorStructure R)
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variable (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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@ -66,8 +87,8 @@ instance : Module R (𝓣.Tensor cX) := PiTensorProduct.instModule
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/-- Equivalence of `ColorModule` given an equality of colors. -/
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def colorModuleCast (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModule ν where
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toFun x := Equiv.cast (congrArg 𝓣.ColorModule h) x
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invFun x := (Equiv.cast (congrArg 𝓣.ColorModule h)).symm x
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toFun := Equiv.cast (congrArg 𝓣.ColorModule h)
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invFun := (Equiv.cast (congrArg 𝓣.ColorModule h)).symm
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map_add' x y := by
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subst h
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rfl
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@ -471,356 +492,6 @@ def decompEmbed (f : Y ↪ X) :
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(𝓣.mapIso (decompEmbedSet f) (𝓣.decompEmbed_cond cX f)) ≪≫ₗ
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(𝓣.tensoratorEquiv (𝓣.decompEmbedColorLeft cX f) (𝓣.decompEmbedColorRight cX f)).symm
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/-!
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## Contraction
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-/
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/-- A linear map taking tensors mapped with the same index set to the product of paired tensors. -/
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def pairProd : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cX2 →ₗ[R]
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⨂[R] x, 𝓣.ColorModule (cX x) ⊗[R] 𝓣.ColorModule (cX2 x) :=
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TensorProduct.lift (
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PiTensorProduct.map₂ (fun x =>
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TensorProduct.mk R (𝓣.ColorModule (cX x)) (𝓣.ColorModule (cX2 x))))
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lemma mkPiAlgebra_equiv (e : X ≃ Y) :
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(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) =
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(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R Y R)) ∘ₗ
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(PiTensorProduct.reindex R _ e).toLinearMap := by
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
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simp only [LinearMap.compMultilinearMap_apply, PiTensorProduct.lift.tprod,
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MultilinearMap.mkPiAlgebra_apply, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
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PiTensorProduct.reindex_tprod, Equiv.prod_comp]
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/-- Given a tensor in `𝓣.Tensor cX` and a tensor in `𝓣.Tensor (𝓣.τ ∘ cX)`, the element of
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`R` formed by contracting all of their indices. -/
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def contrAll' : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor (𝓣.τ ∘ cX) →ₗ[R] R :=
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(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) ∘ₗ
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(PiTensorProduct.map (fun x => 𝓣.contrDual (cX x))) ∘ₗ
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(𝓣.pairProd)
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lemma contrAll'_mapIso_cond {e : X ≃ Y} (h : cX = cY ∘ e) :
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𝓣.τ ∘ cY = (𝓣.τ ∘ cX) ∘ ⇑e.symm := by
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subst h
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exact (Equiv.eq_comp_symm e (𝓣.τ ∘ cY) (𝓣.τ ∘ cY ∘ ⇑e)).mpr rfl
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@[simp]
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lemma contrAll'_mapIso (e : X ≃ Y) (h : c = cY ∘ e) :
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𝓣.contrAll' ∘ₗ
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(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap =
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𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl R _)
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(𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h))).toLinearMap := by
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apply TensorProduct.ext'
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refine fun x ↦
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PiTensorProduct.induction_on' x ?_ (by
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intro a b hx hy y
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simp [map_add, add_tmul, hx, hy])
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intro rx fx
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refine fun y ↦
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PiTensorProduct.induction_on' y ?_ (by
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intro a b hx hy
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simp at hx hy
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simp [map_add, tmul_add, hx, hy])
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intro ry fy
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simp [contrAll']
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rw [mkPiAlgebra_equiv e]
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apply congrArg
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simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
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apply congrArg
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rw [← LinearEquiv.symm_apply_eq]
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rw [PiTensorProduct.reindex_symm]
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rw [← PiTensorProduct.map_reindex]
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subst h
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simp only [Equiv.symm_symm_apply, Function.comp_apply]
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apply congrArg
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rw [pairProd, pairProd]
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simp only [Function.comp_apply, lift.tmul, LinearMapClass.map_smul, LinearMap.smul_apply]
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apply congrArg
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change _ = ((PiTensorProduct.map₂ fun x => TensorProduct.mk R (𝓣.ColorModule (cY (e x)))
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(𝓣.ColorModule (𝓣.τ (cY (e x)))))
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((PiTensorProduct.tprod R) fx))
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((𝓣.mapIso e.symm _) ((PiTensorProduct.tprod R) fy))
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rw [mapIso_tprod]
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simp only [Equiv.symm_symm_apply, Function.comp_apply]
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rw [PiTensorProduct.map₂_tprod_tprod]
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change PiTensorProduct.reindex R _ e.symm
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((PiTensorProduct.map₂ _
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((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (fx (e.symm i))))
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((PiTensorProduct.tprod R) fy)) = _
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rw [PiTensorProduct.map₂_tprod_tprod]
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simp only [Equiv.symm_symm_apply, Function.comp_apply, mk_apply]
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erw [PiTensorProduct.reindex_tprod]
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apply congrArg
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funext i
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simp only [Equiv.symm_symm_apply]
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congr
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simp [colorModuleCast]
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apply cast_eq_iff_heq.mpr
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rw [Equiv.symm_apply_apply]
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@[simp]
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lemma contrAll'_mapIso_tmul (e : X ≃ Y) (h : c = cY ∘ e) (x : 𝓣.Tensor c)
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(y : 𝓣.Tensor (𝓣.τ ∘ cY)) : 𝓣.contrAll' ((𝓣.mapIso e h) x ⊗ₜ[R] y) =
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𝓣.contrAll' (x ⊗ₜ[R] (𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h) y)) := by
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change (𝓣.contrAll' ∘ₗ
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(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap) (x ⊗ₜ[R] y) = _
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rw [contrAll'_mapIso]
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rfl
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/-- The contraction of all the indices of two tensors with dual colors. -/
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def contrAll {c : X → 𝓣.Color} {d : Y → 𝓣.Color}
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(e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d →ₗ[R] R :=
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𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl _ _)
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(𝓣.mapIso e.symm (by subst h; funext a; simp; rw [𝓣.τ_involutive]))).toLinearMap
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lemma contrAll_symm_cond {e : X ≃ Y} (h : c = 𝓣.τ ∘ cY ∘ e) :
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cY = 𝓣.τ ∘ c ∘ ⇑e.symm := by
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subst h
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ext1 x
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simp only [Function.comp_apply, Equiv.apply_symm_apply]
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rw [𝓣.τ_involutive]
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lemma contrAll_mapIso_right_cond {e : X ≃ Y} {e' : Z ≃ Y}
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') : c = 𝓣.τ ∘ cZ ∘ ⇑(e.trans e'.symm) := by
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subst h h'
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ext1 x
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simp only [Function.comp_apply, Equiv.coe_trans, Equiv.apply_symm_apply]
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@[simp]
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lemma contrAll_mapIso_right_tmul (e : X ≃ Y) (e' : Z ≃ Y)
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (x : 𝓣.Tensor c) (z : 𝓣.Tensor cZ) :
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𝓣.contrAll e h (x ⊗ₜ[R] 𝓣.mapIso e' h' z) =
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𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') (x ⊗ₜ[R] z) := by
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rw [contrAll, contrAll]
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simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, congr_tmul,
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LinearEquiv.refl_apply, mapIso_mapIso]
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congr
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@[simp]
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lemma contrAll_comp_mapIso_right (e : X ≃ Y) (e' : Z ≃ Y)
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') : 𝓣.contrAll e h ∘ₗ
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(TensorProduct.congr (LinearEquiv.refl R (𝓣.Tensor c)) (𝓣.mapIso e' h')).toLinearMap
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= 𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') := by
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apply TensorProduct.ext'
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intro x y
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exact 𝓣.contrAll_mapIso_right_tmul e e' h h' x y
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lemma contrAll_mapIso_left_cond {e : X ≃ Y} {e' : Z ≃ X}
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') : cZ = 𝓣.τ ∘ cY ∘ ⇑(e'.trans e) := by
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subst h h'
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ext1 x
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simp only [Function.comp_apply, Equiv.coe_trans, Equiv.apply_symm_apply]
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@[simp]
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lemma contrAll_mapIso_left_tmul {e : X ≃ Y} {e' : Z ≃ X}
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') (x : 𝓣.Tensor cZ) (y : 𝓣.Tensor cY) :
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𝓣.contrAll e h (𝓣.mapIso e' h' x ⊗ₜ[R] y) =
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𝓣.contrAll (e'.trans e) (𝓣.contrAll_mapIso_left_cond h h') (x ⊗ₜ[R] y) := by
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rw [contrAll, contrAll]
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simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, congr_tmul,
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LinearEquiv.refl_apply, contrAll'_mapIso_tmul, mapIso_mapIso]
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congr
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@[simp]
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lemma contrAll_mapIso_left {e : X ≃ Y} {e' : Z ≃ X}
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(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') :
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𝓣.contrAll e h ∘ₗ
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(TensorProduct.congr (𝓣.mapIso e' h') (LinearEquiv.refl R (𝓣.Tensor cY))).toLinearMap
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= 𝓣.contrAll (e'.trans e) (𝓣.contrAll_mapIso_left_cond h h') := by
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apply TensorProduct.ext'
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intro x y
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exact 𝓣.contrAll_mapIso_left_tmul h h' x y
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end PreTensorStructure
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/-! TODO: Add unit here. -/
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/-- A `PreTensorStructure` with the additional constraint that `contrDua` is symmetric. -/
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structure TensorStructure (R : Type) [CommSemiring R] extends PreTensorStructure R where
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/-- The symmetry condition on `contrDua`. -/
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contrDual_symm : ∀ μ,
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(contrDual μ) ∘ₗ (TensorProduct.comm R (ColorModule (τ μ)) (ColorModule μ)).toLinearMap =
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(contrDual (τ μ)) ∘ₗ (TensorProduct.congr (LinearEquiv.refl _ _)
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(toPreTensorStructure.colorModuleCast (by rw[toPreTensorStructure.τ_involutive]))).toLinearMap
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namespace TensorStructure
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open PreTensorStructure
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variable (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
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{c c₂ : X → 𝓣.Color} {d : Y → 𝓣.Color} {b : Z → 𝓣.Color} {d' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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end TensorStructure
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/-- A `TensorStructure` with a group action. -/
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structure GroupTensorStructure (R : Type) [CommSemiring R]
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(G : Type) [Group G] extends TensorStructure R where
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/-- For each color `μ` a representation of `G` on `ColorModule μ`. -/
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repColorModule : (μ : Color) → Representation R G (ColorModule μ)
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/-- The contraction of a vector with its dual is invariant under the group action. -/
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contrDual_inv : ∀ μ g, contrDual μ ∘ₗ
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TensorProduct.map (repColorModule μ g) (repColorModule (τ μ) g) = contrDual μ
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namespace GroupTensorStructure
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open TensorStructure
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open PreTensorStructure
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variable {G : Type} [Group G]
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variable (𝓣 : GroupTensorStructure R G)
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variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
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{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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/-- The representation of the group `G` on the vector space of tensors. -/
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def rep : Representation R G (𝓣.Tensor cX) where
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toFun g := PiTensorProduct.map (fun x => 𝓣.repColorModule (cX x) g)
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map_one' := by
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simp_all only [_root_.map_one, PiTensorProduct.map_one]
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map_mul' g g' := by
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simp_all only [_root_.map_mul]
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exact PiTensorProduct.map_mul _ _
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local infixl:78 " • " => 𝓣.rep
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lemma repColorModule_colorModuleCast_apply (h : μ = ν) (g : G) (x : 𝓣.ColorModule μ) :
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(𝓣.repColorModule ν g) (𝓣.colorModuleCast h x) =
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(𝓣.colorModuleCast h) (𝓣.repColorModule μ g x) := by
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subst h
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simp [colorModuleCast]
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@[simp]
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lemma repColorModule_colorModuleCast (h : μ = ν) (g : G) :
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(𝓣.repColorModule ν g) ∘ₗ (𝓣.colorModuleCast h).toLinearMap =
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(𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
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apply LinearMap.ext
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intro x
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simp [repColorModule_colorModuleCast_apply]
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@[simp]
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lemma rep_mapIso (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) :
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(𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap = (𝓣.mapIso e h).toLinearMap ∘ₗ 𝓣.rep g := by
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apply PiTensorProduct.ext
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apply MultilinearMap.ext
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intro x
|
||||
simp only [LinearMap.compMultilinearMap_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
|
||||
Function.comp_apply]
|
||||
erw [mapIso_tprod]
|
||||
simp [rep, repColorModule_colorModuleCast_apply]
|
||||
change (PiTensorProduct.map fun x => (𝓣.repColorModule (cY x)) g)
|
||||
((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (x (e.symm i))) =
|
||||
(𝓣.mapIso e h) ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) x))
|
||||
rw [PiTensorProduct.map_tprod, PiTensorProduct.map_tprod]
|
||||
rw [mapIso_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
subst h
|
||||
simp [repColorModule_colorModuleCast_apply]
|
||||
|
||||
@[simp]
|
||||
lemma rep_mapIso_apply (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) (x : 𝓣.Tensor cX) :
|
||||
g • (𝓣.mapIso e h x) = (𝓣.mapIso e h) (g • x) := by
|
||||
trans ((𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap) x
|
||||
rfl
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (cX i)) :
|
||||
g • (PiTensorProduct.tprod R f) = PiTensorProduct.tprod R (fun x =>
|
||||
𝓣.repColorModule (cX x) g (f x)) := by
|
||||
simp [rep]
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.tprod R) f) = _
|
||||
rw [PiTensorProduct.map_tprod]
|
||||
|
||||
/-!
|
||||
|
||||
## Group acting on tensor products
|
||||
|
||||
-/
|
||||
|
||||
lemma rep_tensoratorEquiv (g : G) :
|
||||
(𝓣.tensoratorEquiv cX cY) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.rep g ∘ₗ
|
||||
(𝓣.tensoratorEquiv cX cY).toLinearMap := by
|
||||
apply tensorProd_piTensorProd_ext
|
||||
intro p q
|
||||
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, map_tmul, rep_tprod,
|
||||
tensoratorEquiv_tmul_tprod]
|
||||
apply congrArg
|
||||
funext x
|
||||
match x with
|
||||
| Sum.inl x => rfl
|
||||
| Sum.inr x => rfl
|
||||
|
||||
lemma rep_tensoratorEquiv_apply (g : G) (x : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY) :
|
||||
(𝓣.tensoratorEquiv cX cY) ((TensorProduct.map (𝓣.rep g) (𝓣.rep g)) x)
|
||||
= (𝓣.rep g) ((𝓣.tensoratorEquiv cX cY) x) := by
|
||||
trans ((𝓣.tensoratorEquiv cX cY) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x
|
||||
rfl
|
||||
rw [rep_tensoratorEquiv]
|
||||
rfl
|
||||
|
||||
lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor cX) (y : 𝓣.Tensor cY) :
|
||||
(𝓣.tensoratorEquiv cX cY) ((g • x) ⊗ₜ[R] (g • y)) =
|
||||
g • ((𝓣.tensoratorEquiv cX cY) (x ⊗ₜ[R] y)) := by
|
||||
nth_rewrite 1 [← rep_tensoratorEquiv_apply]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Group acting on contraction
|
||||
|
||||
-/
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) (g : G) :
|
||||
𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.contrAll e h := by
|
||||
apply TensorProduct.ext'
|
||||
refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
|
||||
intro a b hx hy y
|
||||
simp [map_add, add_tmul, hx, hy])
|
||||
intro rx fx
|
||||
refine fun y ↦ PiTensorProduct.induction_on' y ?_ (by
|
||||
intro a b hx hy
|
||||
simp at hx hy
|
||||
simp [map_add, tmul_add, hx, hy])
|
||||
intro ry fy
|
||||
simp [contrAll, TensorProduct.smul_tmul]
|
||||
apply congrArg
|
||||
apply congrArg
|
||||
simp [contrAll']
|
||||
apply congrArg
|
||||
simp [pairProd]
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
|
||||
(PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
|
||||
rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
|
||||
PiTensorProduct.map_tprod]
|
||||
simp only [mk_apply]
|
||||
apply congrArg
|
||||
funext x
|
||||
rw [← repColorModule_colorModuleCast_apply]
|
||||
nth_rewrite 2 [← 𝓣.contrDual_inv (c x) g]
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
|
||||
(g : G) (x : 𝓣.Tensor c ⊗ 𝓣.Tensor d) :
|
||||
𝓣.contrAll e h (TensorProduct.map (𝓣.rep g) (𝓣.rep g) x) = 𝓣.contrAll e h x := by
|
||||
change (𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x = _
|
||||
rw [contrAll_rep]
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
|
||||
(g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
|
||||
𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
|
||||
nth_rewrite 2 [← contrAll_rep_apply]
|
||||
rfl
|
||||
|
||||
end GroupTensorStructure
|
||||
|
||||
end
|
||||
|
|
|
|||
201
HepLean/SpaceTime/LorentzTensor/Contractions.lean
Normal file
201
HepLean/SpaceTime/LorentzTensor/Contractions.lean
Normal file
|
|
@ -0,0 +1,201 @@
|
|||
/-
|
||||
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.SpaceTime.LorentzTensor.Basic
|
||||
/-!
|
||||
|
||||
# Contraction of indices
|
||||
|
||||
-/
|
||||
noncomputable section
|
||||
|
||||
open TensorProduct
|
||||
|
||||
variable {R : Type} [CommSemiring R]
|
||||
|
||||
namespace TensorStructure
|
||||
|
||||
variable (𝓣 : TensorStructure R)
|
||||
|
||||
variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
|
||||
{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
|
||||
{cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
|
||||
|
||||
/-- The contraction of a vector in `𝓣.ColorModule ν` with a vector in
|
||||
`𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in `𝓣.ColorModule η`. -/
|
||||
def contrOneTwo {ν η : 𝓣.Color} :
|
||||
𝓣.ColorModule ν ⊗[R] 𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η →ₗ[R] 𝓣.ColorModule η :=
|
||||
(TensorProduct.lid R _).toLinearMap ∘ₗ
|
||||
TensorProduct.map (𝓣.contrDual ν) (LinearEquiv.refl R (𝓣.ColorModule η)).toLinearMap
|
||||
∘ₗ (TensorProduct.assoc R _ _ _).symm.toLinearMap
|
||||
|
||||
/-- The contraction of a vector in `𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule ν` with a vector in
|
||||
`𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in
|
||||
`𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule η`. -/
|
||||
def contrTwoTwo {μ ν η : 𝓣.Color} :
|
||||
(𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule ν) ⊗[R] (𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η) →ₗ[R]
|
||||
𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule η :=
|
||||
(TensorProduct.map (LinearEquiv.refl R _).toLinearMap (𝓣.contrOneTwo)) ∘ₗ
|
||||
(TensorProduct.assoc R _ _ _).toLinearMap
|
||||
|
||||
/-- A linear map taking tensors mapped with the same index set to the product of paired tensors. -/
|
||||
def pairProd : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cX2 →ₗ[R]
|
||||
⨂[R] x, 𝓣.ColorModule (cX x) ⊗[R] 𝓣.ColorModule (cX2 x) :=
|
||||
TensorProduct.lift (
|
||||
PiTensorProduct.map₂ (fun x =>
|
||||
TensorProduct.mk R (𝓣.ColorModule (cX x)) (𝓣.ColorModule (cX2 x))))
|
||||
|
||||
lemma mkPiAlgebra_equiv (e : X ≃ Y) :
|
||||
(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) =
|
||||
(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R Y R)) ∘ₗ
|
||||
(PiTensorProduct.reindex R _ e).toLinearMap := by
|
||||
apply PiTensorProduct.ext
|
||||
apply MultilinearMap.ext
|
||||
intro x
|
||||
simp only [LinearMap.compMultilinearMap_apply, PiTensorProduct.lift.tprod,
|
||||
MultilinearMap.mkPiAlgebra_apply, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
|
||||
PiTensorProduct.reindex_tprod, Equiv.prod_comp]
|
||||
|
||||
/-- Given a tensor in `𝓣.Tensor cX` and a tensor in `𝓣.Tensor (𝓣.τ ∘ cX)`, the element of
|
||||
`R` formed by contracting all of their indices. -/
|
||||
def contrAll' : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor (𝓣.τ ∘ cX) →ₗ[R] R :=
|
||||
(PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R)) ∘ₗ
|
||||
(PiTensorProduct.map (fun x => 𝓣.contrDual (cX x))) ∘ₗ
|
||||
(𝓣.pairProd)
|
||||
|
||||
lemma contrAll'_mapIso_cond {e : X ≃ Y} (h : cX = cY ∘ e) :
|
||||
𝓣.τ ∘ cY = (𝓣.τ ∘ cX) ∘ ⇑e.symm := by
|
||||
subst h
|
||||
exact (Equiv.eq_comp_symm e (𝓣.τ ∘ cY) (𝓣.τ ∘ cY ∘ ⇑e)).mpr rfl
|
||||
|
||||
@[simp]
|
||||
lemma contrAll'_mapIso (e : X ≃ Y) (h : c = cY ∘ e) :
|
||||
𝓣.contrAll' ∘ₗ
|
||||
(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap =
|
||||
𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl R _)
|
||||
(𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h))).toLinearMap := by
|
||||
apply TensorProduct.ext'
|
||||
refine fun x ↦
|
||||
PiTensorProduct.induction_on' x ?_ (by
|
||||
intro a b hx hy y
|
||||
simp [map_add, add_tmul, hx, hy])
|
||||
intro rx fx
|
||||
refine fun y ↦
|
||||
PiTensorProduct.induction_on' y ?_ (by
|
||||
intro a b hx hy
|
||||
simp at hx hy
|
||||
simp [map_add, tmul_add, hx, hy])
|
||||
intro ry fy
|
||||
simp [contrAll']
|
||||
rw [mkPiAlgebra_equiv e]
|
||||
apply congrArg
|
||||
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
|
||||
apply congrArg
|
||||
rw [← LinearEquiv.symm_apply_eq]
|
||||
rw [PiTensorProduct.reindex_symm]
|
||||
rw [← PiTensorProduct.map_reindex]
|
||||
subst h
|
||||
simp only [Equiv.symm_symm_apply, Function.comp_apply]
|
||||
apply congrArg
|
||||
rw [pairProd, pairProd]
|
||||
simp only [Function.comp_apply, lift.tmul, LinearMapClass.map_smul, LinearMap.smul_apply]
|
||||
apply congrArg
|
||||
change _ = ((PiTensorProduct.map₂ fun x => TensorProduct.mk R (𝓣.ColorModule (cY (e x)))
|
||||
(𝓣.ColorModule (𝓣.τ (cY (e x)))))
|
||||
((PiTensorProduct.tprod R) fx))
|
||||
((𝓣.mapIso e.symm _) ((PiTensorProduct.tprod R) fy))
|
||||
rw [mapIso_tprod]
|
||||
simp only [Equiv.symm_symm_apply, Function.comp_apply]
|
||||
rw [PiTensorProduct.map₂_tprod_tprod]
|
||||
change PiTensorProduct.reindex R _ e.symm
|
||||
((PiTensorProduct.map₂ _
|
||||
((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (fx (e.symm i))))
|
||||
((PiTensorProduct.tprod R) fy)) = _
|
||||
rw [PiTensorProduct.map₂_tprod_tprod]
|
||||
simp only [Equiv.symm_symm_apply, Function.comp_apply, mk_apply]
|
||||
erw [PiTensorProduct.reindex_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
simp only [Equiv.symm_symm_apply]
|
||||
congr
|
||||
simp [colorModuleCast]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.symm_apply_apply]
|
||||
|
||||
@[simp]
|
||||
lemma contrAll'_mapIso_tmul (e : X ≃ Y) (h : c = cY ∘ e) (x : 𝓣.Tensor c)
|
||||
(y : 𝓣.Tensor (𝓣.τ ∘ cY)) : 𝓣.contrAll' ((𝓣.mapIso e h) x ⊗ₜ[R] y) =
|
||||
𝓣.contrAll' (x ⊗ₜ[R] (𝓣.mapIso e.symm (𝓣.contrAll'_mapIso_cond h) y)) := by
|
||||
change (𝓣.contrAll' ∘ₗ
|
||||
(TensorProduct.congr (𝓣.mapIso e h) (LinearEquiv.refl R _)).toLinearMap) (x ⊗ₜ[R] y) = _
|
||||
rw [contrAll'_mapIso]
|
||||
rfl
|
||||
|
||||
/-- The contraction of all the indices of two tensors with dual colors. -/
|
||||
def contrAll {c : X → 𝓣.Color} {d : Y → 𝓣.Color}
|
||||
(e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) : 𝓣.Tensor c ⊗[R] 𝓣.Tensor d →ₗ[R] R :=
|
||||
𝓣.contrAll' ∘ₗ (TensorProduct.congr (LinearEquiv.refl _ _)
|
||||
(𝓣.mapIso e.symm (by funext a; simpa [h] using (𝓣.τ_involutive _).symm))).toLinearMap
|
||||
|
||||
lemma contrAll_symm_cond {e : X ≃ Y} (h : c = 𝓣.τ ∘ cY ∘ e) :
|
||||
cY = 𝓣.τ ∘ c ∘ ⇑e.symm := by
|
||||
subst h
|
||||
ext1 x
|
||||
simp only [Function.comp_apply, Equiv.apply_symm_apply]
|
||||
rw [𝓣.τ_involutive]
|
||||
|
||||
lemma contrAll_mapIso_right_cond {e : X ≃ Y} {e' : Z ≃ Y}
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') : c = 𝓣.τ ∘ cZ ∘ ⇑(e.trans e'.symm) := by
|
||||
subst h h'
|
||||
ext1 x
|
||||
simp only [Function.comp_apply, Equiv.coe_trans, Equiv.apply_symm_apply]
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_mapIso_right_tmul (e : X ≃ Y) (e' : Z ≃ Y)
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') (x : 𝓣.Tensor c) (z : 𝓣.Tensor cZ) :
|
||||
𝓣.contrAll e h (x ⊗ₜ[R] 𝓣.mapIso e' h' z) =
|
||||
𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') (x ⊗ₜ[R] z) := by
|
||||
rw [contrAll, contrAll]
|
||||
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, congr_tmul,
|
||||
LinearEquiv.refl_apply, mapIso_mapIso]
|
||||
congr
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_comp_mapIso_right (e : X ≃ Y) (e' : Z ≃ Y)
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = cY ∘ e') : 𝓣.contrAll e h ∘ₗ
|
||||
(TensorProduct.congr (LinearEquiv.refl R (𝓣.Tensor c)) (𝓣.mapIso e' h')).toLinearMap
|
||||
= 𝓣.contrAll (e.trans e'.symm) (𝓣.contrAll_mapIso_right_cond h h') := by
|
||||
apply TensorProduct.ext'
|
||||
intro x y
|
||||
exact 𝓣.contrAll_mapIso_right_tmul e e' h h' x y
|
||||
|
||||
lemma contrAll_mapIso_left_cond {e : X ≃ Y} {e' : Z ≃ X}
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') : cZ = 𝓣.τ ∘ cY ∘ ⇑(e'.trans e) := by
|
||||
subst h h'
|
||||
ext1 x
|
||||
simp only [Function.comp_apply, Equiv.coe_trans, Equiv.apply_symm_apply]
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_mapIso_left_tmul {e : X ≃ Y} {e' : Z ≃ X}
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') (x : 𝓣.Tensor cZ) (y : 𝓣.Tensor cY) :
|
||||
𝓣.contrAll e h (𝓣.mapIso e' h' x ⊗ₜ[R] y) =
|
||||
𝓣.contrAll (e'.trans e) (𝓣.contrAll_mapIso_left_cond h h') (x ⊗ₜ[R] y) := by
|
||||
rw [contrAll, contrAll]
|
||||
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, congr_tmul,
|
||||
LinearEquiv.refl_apply, contrAll'_mapIso_tmul, mapIso_mapIso]
|
||||
congr
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_mapIso_left {e : X ≃ Y} {e' : Z ≃ X}
|
||||
(h : c = 𝓣.τ ∘ cY ∘ e) (h' : cZ = c ∘ e') :
|
||||
𝓣.contrAll e h ∘ₗ
|
||||
(TensorProduct.congr (𝓣.mapIso e' h') (LinearEquiv.refl R (𝓣.Tensor cY))).toLinearMap
|
||||
= 𝓣.contrAll (e'.trans e) (𝓣.contrAll_mapIso_left_cond h h') := by
|
||||
apply TensorProduct.ext'
|
||||
intro x y
|
||||
exact 𝓣.contrAll_mapIso_left_tmul h h' x y
|
||||
|
||||
end TensorStructure
|
||||
20
HepLean/SpaceTime/LorentzTensor/Fin.lean
Normal file
20
HepLean/SpaceTime/LorentzTensor/Fin.lean
Normal file
|
|
@ -0,0 +1,20 @@
|
|||
/-
|
||||
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.SpaceTime.LorentzTensor.Basic
|
||||
/-!
|
||||
|
||||
# Lorentz tensors indexed by Fin n
|
||||
|
||||
In physics, in many (if not all) cases, we index our Lorentz tensors by `Fin n`.
|
||||
|
||||
In this file we set up the machinery to deal with Lorentz tensors indexed by `Fin n`
|
||||
in a way that is convenient for physicists, and general caculation.
|
||||
|
||||
## Note
|
||||
|
||||
This file is currently a stub.
|
||||
|
||||
-/
|
||||
185
HepLean/SpaceTime/LorentzTensor/LorentzTensorStruct.lean
Normal file
185
HepLean/SpaceTime/LorentzTensor/LorentzTensorStruct.lean
Normal file
|
|
@ -0,0 +1,185 @@
|
|||
/-
|
||||
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.SpaceTime.LorentzTensor.RisingLowering
|
||||
import Mathlib.RepresentationTheory.Basic
|
||||
/-!
|
||||
|
||||
# Structure to form Lorentz-style Tensor
|
||||
|
||||
-/
|
||||
|
||||
noncomputable section
|
||||
|
||||
open TensorProduct
|
||||
|
||||
variable {R : Type} [CommSemiring R]
|
||||
|
||||
/-! TODO: Add preservation of the unit, and the metric. -/
|
||||
/-- A `DualizeTensorStructure` with a group action. -/
|
||||
structure LorentzTensorStructure (R : Type) [CommSemiring R]
|
||||
(G : Type) [Group G] extends DualizeTensorStructure R where
|
||||
/-- For each color `μ` a representation of `G` on `ColorModule μ`. -/
|
||||
repColorModule : (μ : Color) → Representation R G (ColorModule μ)
|
||||
/-- The contraction of a vector with its dual is invariant under the group action. -/
|
||||
contrDual_inv : ∀ μ g, contrDual μ ∘ₗ
|
||||
TensorProduct.map (repColorModule μ g) (repColorModule (τ μ) g) = contrDual μ
|
||||
|
||||
namespace LorentzTensorStructure
|
||||
open TensorStructure
|
||||
|
||||
variable {G : Type} [Group G]
|
||||
|
||||
variable (𝓣 : LorentzTensorStructure R G)
|
||||
|
||||
variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
|
||||
[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
|
||||
{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
|
||||
|
||||
/-- The representation of the group `G` on the vector space of tensors. -/
|
||||
def rep : Representation R G (𝓣.Tensor cX) where
|
||||
toFun g := PiTensorProduct.map (fun x => 𝓣.repColorModule (cX x) g)
|
||||
map_one' := by
|
||||
simp_all only [_root_.map_one, PiTensorProduct.map_one]
|
||||
map_mul' g g' := by
|
||||
simp_all only [_root_.map_mul]
|
||||
exact PiTensorProduct.map_mul _ _
|
||||
|
||||
local infixl:78 " • " => 𝓣.rep
|
||||
|
||||
lemma repColorModule_colorModuleCast_apply (h : μ = ν) (g : G) (x : 𝓣.ColorModule μ) :
|
||||
(𝓣.repColorModule ν g) (𝓣.colorModuleCast h x) =
|
||||
(𝓣.colorModuleCast h) (𝓣.repColorModule μ g x) := by
|
||||
subst h
|
||||
simp [colorModuleCast]
|
||||
|
||||
@[simp]
|
||||
lemma repColorModule_colorModuleCast (h : μ = ν) (g : G) :
|
||||
(𝓣.repColorModule ν g) ∘ₗ (𝓣.colorModuleCast h).toLinearMap =
|
||||
(𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
|
||||
apply LinearMap.ext
|
||||
intro x
|
||||
simp [repColorModule_colorModuleCast_apply]
|
||||
|
||||
@[simp]
|
||||
lemma rep_mapIso (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) :
|
||||
(𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap = (𝓣.mapIso e h).toLinearMap ∘ₗ 𝓣.rep g := by
|
||||
apply PiTensorProduct.ext
|
||||
apply MultilinearMap.ext
|
||||
intro x
|
||||
simp only [LinearMap.compMultilinearMap_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
|
||||
Function.comp_apply]
|
||||
erw [mapIso_tprod]
|
||||
simp [rep, repColorModule_colorModuleCast_apply]
|
||||
change (PiTensorProduct.map fun x => (𝓣.repColorModule (cY x)) g)
|
||||
((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (x (e.symm i))) =
|
||||
(𝓣.mapIso e h) ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) x))
|
||||
rw [PiTensorProduct.map_tprod, PiTensorProduct.map_tprod, mapIso_tprod]
|
||||
apply congrArg
|
||||
funext i
|
||||
subst h
|
||||
simp [repColorModule_colorModuleCast_apply]
|
||||
|
||||
@[simp]
|
||||
lemma rep_mapIso_apply (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) (x : 𝓣.Tensor cX) :
|
||||
g • (𝓣.mapIso e h x) = (𝓣.mapIso e h) (g • x) := by
|
||||
trans ((𝓣.rep g) ∘ₗ (𝓣.mapIso e h).toLinearMap) x
|
||||
rfl
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (cX i)) :
|
||||
g • (PiTensorProduct.tprod R f) = PiTensorProduct.tprod R (fun x =>
|
||||
𝓣.repColorModule (cX x) g (f x)) := by
|
||||
simp [rep]
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.tprod R) f) = _
|
||||
rw [PiTensorProduct.map_tprod]
|
||||
|
||||
/-!
|
||||
|
||||
## Group acting on tensor products
|
||||
|
||||
-/
|
||||
|
||||
lemma rep_tensoratorEquiv (g : G) :
|
||||
(𝓣.tensoratorEquiv cX cY) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.rep g ∘ₗ
|
||||
(𝓣.tensoratorEquiv cX cY).toLinearMap := by
|
||||
apply tensorProd_piTensorProd_ext
|
||||
intro p q
|
||||
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, map_tmul, rep_tprod,
|
||||
tensoratorEquiv_tmul_tprod]
|
||||
apply congrArg
|
||||
funext x
|
||||
match x with
|
||||
| Sum.inl x => rfl
|
||||
| Sum.inr x => rfl
|
||||
|
||||
lemma rep_tensoratorEquiv_apply (g : G) (x : 𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY) :
|
||||
(𝓣.tensoratorEquiv cX cY) ((TensorProduct.map (𝓣.rep g) (𝓣.rep g)) x)
|
||||
= (𝓣.rep g) ((𝓣.tensoratorEquiv cX cY) x) := by
|
||||
trans ((𝓣.tensoratorEquiv cX cY) ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x
|
||||
rfl
|
||||
rw [rep_tensoratorEquiv]
|
||||
rfl
|
||||
|
||||
lemma rep_tensoratorEquiv_tmul (g : G) (x : 𝓣.Tensor cX) (y : 𝓣.Tensor cY) :
|
||||
(𝓣.tensoratorEquiv cX cY) ((g • x) ⊗ₜ[R] (g • y)) =
|
||||
g • ((𝓣.tensoratorEquiv cX cY) (x ⊗ₜ[R] y)) := by
|
||||
nth_rewrite 1 [← rep_tensoratorEquiv_apply]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Group acting on contraction
|
||||
|
||||
-/
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e) (g : G) :
|
||||
𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g)) = 𝓣.contrAll e h := by
|
||||
apply TensorProduct.ext'
|
||||
refine fun x ↦ PiTensorProduct.induction_on' x ?_ (by
|
||||
intro a b hx hy y
|
||||
simp [map_add, add_tmul, hx, hy])
|
||||
intro rx fx
|
||||
refine fun y ↦ PiTensorProduct.induction_on' y ?_ (by
|
||||
intro a b hx hy
|
||||
simp at hx hy
|
||||
simp [map_add, tmul_add, hx, hy])
|
||||
intro ry fy
|
||||
simp [contrAll, TensorProduct.smul_tmul]
|
||||
apply congrArg
|
||||
apply congrArg
|
||||
simp [contrAll']
|
||||
apply congrArg
|
||||
simp [pairProd]
|
||||
change (PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _) =
|
||||
(PiTensorProduct.map _) ((PiTensorProduct.map₂ _ _) _)
|
||||
rw [PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map₂_tprod_tprod, PiTensorProduct.map_tprod,
|
||||
PiTensorProduct.map_tprod]
|
||||
simp only [mk_apply]
|
||||
apply congrArg
|
||||
funext x
|
||||
rw [← repColorModule_colorModuleCast_apply]
|
||||
nth_rewrite 2 [← 𝓣.contrDual_inv (c x) g]
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
|
||||
(g : G) (x : 𝓣.Tensor c ⊗ 𝓣.Tensor d) :
|
||||
𝓣.contrAll e h (TensorProduct.map (𝓣.rep g) (𝓣.rep g) x) = 𝓣.contrAll e h x := by
|
||||
change (𝓣.contrAll e h ∘ₗ (TensorProduct.map (𝓣.rep g) (𝓣.rep g))) x = _
|
||||
rw [contrAll_rep]
|
||||
|
||||
@[simp]
|
||||
lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
|
||||
(g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
|
||||
𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
|
||||
nth_rewrite 2 [← contrAll_rep_apply]
|
||||
rfl
|
||||
|
||||
end LorentzTensorStructure
|
||||
|
||||
end
|
||||
23
HepLean/SpaceTime/LorentzTensor/Notation.lean
Normal file
23
HepLean/SpaceTime/LorentzTensor/Notation.lean
Normal file
|
|
@ -0,0 +1,23 @@
|
|||
/-
|
||||
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.SpaceTime.LorentzTensor.Basic
|
||||
/-!
|
||||
|
||||
# Notation for Lorentz Tensors
|
||||
|
||||
This file is currently a stub.
|
||||
|
||||
We plan to set up index-notation for dealing with tensors.
|
||||
|
||||
Some examples:
|
||||
|
||||
- `ψᵘ¹ᵘ²φᵤ₁` should correspond to the contraction of the first index of `ψ` and the
|
||||
only index of `φ`.
|
||||
- `ψᵘ¹ᵘ² = ψᵘ²ᵘ¹` should define the symmetry of `ψ` under the exchange of its indices.
|
||||
|
||||
It should also be possible to define this generically for any `LorentzTensorStructure`.
|
||||
|
||||
-/
|
||||
29
HepLean/SpaceTime/LorentzTensor/RisingLowering.lean
Normal file
29
HepLean/SpaceTime/LorentzTensor/RisingLowering.lean
Normal file
|
|
@ -0,0 +1,29 @@
|
|||
/-
|
||||
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.SpaceTime.LorentzTensor.Contractions
|
||||
/-!
|
||||
|
||||
# Structure for Rising and Lowering indices
|
||||
|
||||
-/
|
||||
|
||||
noncomputable section
|
||||
|
||||
open TensorProduct
|
||||
|
||||
variable {R : Type} [CommSemiring R]
|
||||
|
||||
/-- Structure extending `TensorStructure` with the addition of a metric
|
||||
permitting to `rise` and `lower` indices. -/
|
||||
structure DualizeTensorStructure (R : Type) [CommSemiring R] extends TensorStructure R where
|
||||
/-- The metric for a given color. -/
|
||||
metric : Color → ColorModule (τ μ) ⊗[R] ColorModule (τ μ)
|
||||
/-- The metric contracted with its dual is the unit. -/
|
||||
metric_dual : ∀ μ,
|
||||
TensorProduct.congr (LinearEquiv.refl _ _) (toTensorStructure.colorModuleCast (τ_involutive μ))
|
||||
(toTensorStructure.contrTwoTwo (metric μ ⊗ₜ[R] metric (τ μ))) = unit μ
|
||||
|
||||
end
|
||||
Loading…
Add table
Add a link
Reference in a new issue